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Advanced Mathematic s Educa tional S upport Recommendations and Resources for Facilitating Collaboration Between Higher Education Mathematics Faculty and Texas Public High Schools First printing August 2003 Copyright 2003, The University of Texas at Austin. All rights reserved. Permission is given to any person, group, or organization to copy and distribute this publication, Advanced Mathematics Educational Support, for noncommercial educational purposes only, so long as the appropriate credit is given. This permission is granted by the Charles A. Dana Center, a unit of the College of Natural Sciences at The University of Texas at Austin. About the Charles A. Dana Center’s Work in Mathematics and Science The Charles A. Dana Center at The University of Texas at Austin works to support education leaders and policymakers in strengthening Texas education. As a research unit of The University of Texas at Austin’s College of Natural Sciences, the Dana Center maintains a special emphasis on mathematics and science education. We offer professional development institutes and produce research-based mathematics and science resources for educators to use in helping all students achieve academic success. For more information, visit the Dana Center website at www.utdanacenter.org. The development of Advanced Mathematics Educational Support was supported in part by the Sid. W. Richardson Foundation and the Charles A. Dana Center at The University of Texas at Austin. Additional funding was provided by an anonymous donor. Any opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the Sid W. Richardson Foundation or The University of Texas at Austin. Author team and acknowledgments Primary Authors Ray Cannon, Department of Mathematics, Baylor University Richard Parr, School Mathematics Project, Rice University Ann Webb, College of Education, University of Texas at Tyler Charles A. Dana Center Production Team Bill Hopkins, AP Equity Initiative, Lead Editor Kathi Cook, AP Equity Initiative, Editor Susan Hudson Hull, Mathematics, Editor Rachel Jenkins, Copy Editor Amy Dolejs, Copy Editor Phil Swann, Communications, Senior Designer Advanced Mathematics Educational Support Advisory Team Jasper Adams Department of Mathematics, Stephen F. Austin State University James Epperson Department of Mathematics, University of Texas at Arlington Gregg Fleisher Advanced Placement Strategies, Inc., Dallas, Texas Tony Hartman Department of Mathematics, Texarkana College Frank Hawkins Department of Mathematics, Prairie View A&M University Barbary Keith Southwestern Regional Office, The College Board Debbie Pace Department of Mathematics, Stephen F. Austin State University Spurgeon Parker Department of Mathematics, Lee High School, Houston Independent School District Vince Schielack Department of Mathematics, Texas A&M University Uri Treisman Charles A. Dana Center and Department of Mathematics, The University of Texas at Austin Gloria White Eisenhower and Teacher Quality Grants, Texas Higher Education Coordinating Board Carol Williams Department of Mathematics, Abilene Christian University Some trademarked designations are used in this publication. Where we were aware of such a designation, the trademarked term has been printed with initial capitalization. Registered trademarks of the College Board used in this publication include Advanced Placement Program, AP, AP Vertical Teams, College Board, Pre-AP, and SAT. (For more information on trademarked terms of The College Board, see “The College Board Trademarks” at www.collegeboard.com/html/trademark001.html.) ACT is a trademark of ACT, Inc. The Texas Academic Skills Program and TASP are trademarks of the Texas Higher Education Coordinating Board and National Evaluation Systems, Inc. ii Advanced Mathematics Educational Support TEKS and TAKS Resources The mathematics Texas Essential Knowledge and Skills (TEKS) were developed by the state of Texas to clarify what all students should know and be able to do in mathematics in kindergarten through grade 12. Districts are required to provide instruction that is aligned with the mathematics TEKS, which were adopted by the State Board of Education in 1997 and implemented statewide in 1998. The mathematics TEKS also constitute the objectives and student expectations for the mathematics portion of the Texas Assessment of Knowledge and Skills (TAKS). The mathematics TEKS can be downloaded in printable format, free of charge, from the Texas Education Agency website (www.tea.state.tx.us/teks). Bound versions of the mathematics and science TEKS are available at cost from the Charles A. Dana Center at The University of Texas at Austin (www.utdanacenter.org or 512-471-6190). Resources for implementing the mathematics TEKS, including educator professional development opportunities, are available through the Texas Education Agency and the Charles A. Dana Center. Online resources can be found in the Mathematics TEKS Toolkit at www.mathtekstoolkit.org. Additional related products and services are available from the Dana Center via www.utdanacenter.org. Advanced Mathematics Educational Support iii Table of Contents Introduction.......................................................................................................................... 1 Why AMES Was Developed.............................................................................................................................................1 How AMES Was Developed.............................................................................................................................................2 Organization of AMES.........................................................................................................................................................2 A Shared Vision for Student Success: K–16 Mathematics in Texas ........................................ 3 The Texas High School Mathematics Program......................................................................................................3 K–12 Accountability in Texas—A Historical Perspective...............................................................................4 TAKS: Increased Accountability for High Schools.............................................................................................5 Transition to College............................................................................................................................................................6 Mathematics Education from the High School Perspective................................................... 7 Teacher Supply and Retention........................................................................................................................................7 The Situation.....................................................................................................................................................................7 Recommendations.........................................................................................................................................................7 Teacher Professional Development and Resources...........................................................................................8 The Situation.....................................................................................................................................................................8 Recommendations.........................................................................................................................................................9 Vertical Teaming....................................................................................................................................................................10 The Situation...................................................................................................................................................................10 Recommendations.......................................................................................................................................................10 Mathematics Education from the Higher Education Perspective.........................................12 Concurrent Enrollment....................................................................................................................................................12 The Situation...................................................................................................................................................................12 Recommendations.......................................................................................................................................................13 Standards for High School Calculus and Statistics.............................................................................................13 The Situation...................................................................................................................................................................13 Recommendations.......................................................................................................................................................14 Tenure and Promotion......................................................................................................................................................15 The Situation...................................................................................................................................................................15 Recommendations.......................................................................................................................................................15 Undergraduate Teacher Preparation..........................................................................................................................16 The Situation...................................................................................................................................................................16 Recommendations.......................................................................................................................................................16 Alternative Teacher Certification.................................................................................................................................17 The Situation...................................................................................................................................................................17 Recommendations.......................................................................................................................................................17 Promising Practices .............................................................................................................18 Master Teacher Summer Institutes..............................................................................................................................18 AP Summer Institute...........................................................................................................................................................18 iv Advanced Mathematics Educational Support The Rice University School Mathematics Project.............................................................................................19 Information Technology in Science Center for Teaching and Learning............................................19 UTeach .....................................................................................................................................................................................20 Systemic/Programmatic Administration Support..............................................................................................21 Lamar State College–Orange Mathematics Institutes........................................................................................22 Appendix A: Chapter 111. Texas Essential Knowledge and Skills for Mathematics .............. 24 Appendix B: Topic Outlines for Advanced Placement Mathematics.................................... 48 AP Calculus AB......................................................................................................................................................................48 AP Calculus BC......................................................................................................................................................................49 AP Statistics...............................................................................................................................................................................50 Appendix C: Vertical Teams: A Strategy for Building School Capacity..................................51 Higher Education Involvement in Vertical Teaming.......................................................................................51 Higher Education Leadership in Vertical Teaming..........................................................................................51 Vertical Teaming Not in Place..............................................................................................................................51 Vertical Teaming Already in Place.....................................................................................................................53 Vertical Alignment of the K–16 Curriculum.......................................................................................................54 References........................................................................................................................... 55 Advanced Mathematics Educational Support v vi Advanced Mathematics Educational Support INTRODUCTION Large-scale and well-replicated national studies (Adelman, 1999; Horn & Carroll, 2001) report extremely high correlations between student participation in a rigorous high school curriculum and their successful and timely completion of postsecondary education. Still, for far too many students, academic preparation in mathematics is focused on passing state tests and completing graduation requirements, rather than on taking challenging academic coursework that would prepare them for success in postsecondary education. This document, Advanced Mathematics Educational Support (AMES), was developed for higher education mathematics faculty to use as they begin their work with schools to address this concern. It builds on the wisdom of promising programs and practices from authors and advisors representing higher education and high schools across Texas. AMES suggests strategies to improve communication and collaboration between high school and higher education faculty, but it is not a synthesis of the research. Instead, it is a collection of information, promising practices, and recommendations designed to spark interest in working with public schools and help higher education faculty begin a useful conversation with their public school counterparts about rigorous coursework and student support. Why AMES Was Developed This document is intended for higher education faculty in mathematics. In it, we attempt to capture the current situation of K–12 mathematics education from the view of public school educators and higher education faculty, explore commonalities in the issues faced by both, and propose some steps that higher education faculty might take to help public school educators meet the goal of preparing all students for success in mathematics after high school. Many college mathematics faculty report a wide variation in the mathematical preparation of their first-year students. Roughly, these students fall into three groups. The first includes the strongest students, those who have successfully completed four years of rigorous mathematics in high school, including Advanced Placement (AP) courses, and who are well-prepared for subsequent mathematics courses. The second group consists of students who have not taken mathematics in their final year of high school and are ill-prepared to succeed in mathematics at the collegiate level. The third group, consisting of students who have taken four years of mathematics courses that were not at sufficient rigor to prepare them for success at the college level, poses the most troublesome problems in placement. To increase the number of students in the first group and decrease the number in the last two requires that college faculty become more directly involved in students’ precollege mathematical preparation. What would it take for public schools to provide a mathematics program so strong and inviting that a large percentage of students—perhaps every student—could be prepared to successfully complete challenging and rigorous mathematics courses throughout their four years in high school? Precollege-level mathematics serves as a gateway to numerous postsecondary and career opportunities for students. A strong mathematics background helps all students develop analytical skills and knowledge that will be valuable in later life in a wide variety of settings. Calculus, in particular, provides a useful background for work in many fields. Students majoring in business, computer science, engineering, and the natural or life sciences in most four-year institutions of higher education are required to successfully complete calculus. Those students must enter college with a foundation in mathematics that at the very least prepares them for calculus. Advanced Mathematics Educational Support 1 Students, parents, high school counselors, teachers, and administrators must be made aware of the fact that—while the state of Texas requires only three years of mathematics for high school graduation—four years of substantive mathematics are needed to prepare for success at the collegiate level. Many higher education mathematics faculty would suggest that more secondary students should build a stronger foundation in algebra, trigonometry, and formal plane geometry—rather than pushing to complete a calculus course in high school. The majority of mathematics teachers on high school campuses would agree. In this document we explore ways to foster dialogue and collaboration between higher education and K–12 faculty, with the ultimate goal of increasing student participation and success in challenging and rigorous mathematics courses throughout four years of high school. How AMES Was Developed AMES is the product of an advisory team working closely with three authors who have experience in building collaborative relationships between their universities and public school teachers. Each member of the advisory team was selected for his or her knowledge of the public school situation or the higher education setting, or for experience in building bridges between the two. AMES is not a comprehensive inventory of strategies, nor is it a research study, or a survey of the literature related to higher education–public school partnerships. Instead, it is a collection of information, promising practices, and recommendations designed to spark interest in working with public schools and help higher education faculty begin a useful conversation with their public school counterparts. Organization of AMES AMES was developed to present views of Texas mathematics education from high school and higher education perspectives. A shared understanding of both will be essential to improve K–16 mathematics in Texas. The section entitled Mathematics Education from the High School Perspective addresses issues including the teaching force, professional development and resources, vertical teams, and district concerns. The Texas accountability system sets the context for the high school perspective. Mathematics Education from the Higher Education Perspective addresses issues of concurrent enrollment, standards for high school calculus and statistics, tenure and promotion, undergraduate teacher preparation, and alternative certification. Each of these sections addresses the situation and includes general recommendations. Promising Practices includes samples of model programs and approaches that address selected issues from the previous two sections. The appendices include related information and resources. 2 Advanced Mathematics Educational Support A SHARED VISION FOR STUDENT SUCCESS: K–16 MATHEMATICS IN TEXAS College mathematics faculty, college mathematics education faculty, and high school mathematics educators share an ambitious vision for student achievement in mathematics. All groups believe that all students should successfully complete high-quality mathematics courses in high school that will enable them to enter higher education prepared to function on-level in mathematics or to enter the workplace with the mathematical preparation necessary to succeed. Thus, in practical terms, in Texas high schools, all students should successfully complete Algebra I, Algebra II, and Geometry, and large percentages of students should successfully complete Precalculus, AP Statistics, and/or AP Calculus. Thus, all middle school, high school, and higher education mathematics educators should be prepared in the mathematics content, pedagogy, and technology that will enable them to deliver high-quality mathematics instruction to students. All public school and higher education mathematics educators should have a good understanding of the state’s standards and vertical alignment for mathematics content K–16, and should work closely together to promote student success. While the state’s standards articulate the vertical alignment in the K–12 mathematics curriculum, there is no such clear alignment between high school and college mathematics. Before effective collaboration between college faculty and high school mathematics educators can take place, both groups must have an understanding of the challenges that their colleagues face daily. The Texas High School Mathematics Program For high school mathematics teachers in Texas, the challenges are many. These challenges include • rigorous curriculum standards defined by the state, • state-mandated testing, • district-mandated curricular changes, • increased pressure to teach more students more mathematics, • changes in the format of the instructional day, • increased attention to meeting the affective needs of students, and • emphasis on educating all students, including those with special needs. In Texas, the high school mathematics program has undergone substantial changes over a short period. In 1997–98, the state adopted the Texas Essential Knowledge and Skills (TEKS)—curriculum guidelines that establish what every student, from elementary school through high school, must know and be able to do in core content areas. For all high school mathematics courses, the TEKS call for increased emphasis on conceptual understanding balanced with student mastery of algorithmic processes and skills. In addition the TEKS call for the appropriate use of technology in all facets of mathematics instruction. Just as the nature of the content areas that students are required to know has changed, so has the definition of what Advanced Mathematics Educational Support 3 those content areas include and the methods that are most effective in teaching those content areas. The TEKS define high school mathematics content for Algebra I, Geometry, Algebra II, Precalculus, and Mathematical Models with Applications. In addition, students may earn state mathematics credit for Advanced Placement Calculus (AB and BC), Advanced Placement Statistics, International Baccalaureate courses, independent studies courses above the Algebra II level, and concurrent enrollment1 courses. The state standards do not give high school credit for courses below the Algebra I level—such as Pre-Algebra, Consumer Mathematics, and Fundamentals of Mathematics—that had been offered in the past. Districts still have the option of offering such mathematics courses for local credit. (Refer to Appendix A for the high school Texas Essential Knowledge and Skills for Mathematics.) To graduate, all Texas students must take three years of high school mathematics, including Algebra I and Geometry. Students in the state’s Recommended High School Program are also required to take Algebra II. Under the Distinguished Academic Program,2 the most rigorous graduation program, students are encouraged, but not required, to take four years of mathematics, including Precalculus. (Compare these mathematics requirements to four years of English Language Arts and Reading required under the state’s minimum graduation plan with options for taking AP English after completing English II.) Also, some schools are currently requiring four years of high school mathematics, beginning with Algebra I, for high school graduation. Beginning with the 2005–06 school year, with the statewide implementation of the Recommended High School Program as mandatory, Texas students will be required to complete three high school mathematics courses through Algebra II. While an improvement, in that it will require students to advance further in their mathematical preparation than is currently required under the minimum program, this new policy will not provide enough incentive to increase enrollments in advanced courses beyond Algebra II. In addition, with the implementation of this new graduation plan, superintendents will struggle even more to find teachers with the content knowledge and pedagogical experience to teach Algebra II.3 K–12 Accountability in Texas—A Historical Perspective Perhaps the issue that most affects classroom mathematics instruction in Texas high schools is state-mandated assessments and their use as graduation requirements and as key components in the Texas educational accountability system. Texas is in a period of transition in its assessment system. Through the 2001–02 school year, the Texas Assessment of Academic Skills (TAAS) was administered to all students in grades 3–8 and in grade 10. Passing both the English Language Arts and Mathematics portions of the 10th-grade TAAS was a state-mandated graduation requirement. 1 The practice of having high school students enroll in college courses while still in high school is known as concurrent or dual enrollment. For more on concurrent enrollment, see “Concurrent Enrollment” in the section headed Mathematics Education from the High School Perspective. 2 For more information on the Recommended High School Program and the Distinguished Academic Program, see www.tea.state.tx.us/rules/tac. 3 Data provided in 2003 from the State Board for Educator Certification indicated that 27.8% of Texas Algebra II teachers were either teaching out of field or were not fully certified (unpublished communication). 4 Advanced Mathematics Educational Support In 2003, the new Texas Assessment of Knowledge and Skills (TAKS) replaces TAAS. TAKS assessments will be given in grades 3–11, with the 11th-grade exam serving as the new exitlevel graduation requirement.4 How students perform on these examinations is important to schools and school districts because student achievement data is used in the Texas School Accountability System. The total percentages of students, including disaggregated subpopulations of students, at a school or district who pass each part of the exam, constitutes a large part of a school or district’s accountability rating. Schools are rated as Exemplary, Recognized, Acceptable, or Low Performing based in part on the passing rates of students on these exams.5 The rating received carries serious implications for the district; repeated low ratings can lead to state sanctions for low-performing districts. In addition, community pressure for schools to reach the higher rating levels may be intense; this in turn leads to great pressure on high school teachers to prepare students to do well on these exams. In the past, the problem with preparing high school students for the state assessments has been the lack of alignment between what was tested on the exit-level TAAS exam given at grade 10 (that is, mathematics through grade 8) and the high school mathematics curriculum. Many schools responded to this pressure in inappropriate ways; for example, some schools conducted intensive review of middle school mathematics during the sophomore mathematics courses, while others instituted programs in which all students participated in TAAS drill and practice. Much classroom time was also devoted to administering TAAS pretests and quizzes that were intended to determine which objectives needed special attention. This meant that sufficient classroom time was not devoted to high school–level mathematics. TAKS: Increased Accountability for High Schools Under the Texas Assessment of Knowledge and Skills system, students will be tested in mathematics in grades 9 and 10 as well as on the exit-level 11th-grade test. The results of all three assessments will be used as a factor in determining campus accountability ratings. More importantly, the new TAKS exam will be better aligned with the high school mathematics TEKS, assessing for the first time student success on Algebra I and Geometry content. In all grades the TEKS and TAKS are more tightly aligned than were the TEKS and the Texas Assessment of Academic Skills, or TAAS. Also, TAKS test items will reflect the increasing rigor begun with the 2000 TAAS test, calling for students to reason and justify as well as demonstrate knowledge and skills as required in the TEKS.6 This means that drill and practice for the TAKS will be unlikely to ensure student success; what matters is that teachers teach students to know and be able to do the mathematics as described in the TEKS. Another change is that the high school mathematics TAKS will require that students have access to a graphing calculator in order to answer questions analyzing multiple mathematics relationships and representations. 4 The exit-level TAKS assesses student performance in mathematics, English language arts, social studies, and science. 5 For a brief introduction to this accountability system, go to “Accountability,” at www.tea.state.tx.us/accountability.html. For a more detailed explanation of the new system, see Texas Education Agency, Department of Accountability Reporting and Research, 2003 Accountability Plan, available at www.tea.state.tx.us/perfreport/account/2003/plan/index.html. 6 Samples of TAKS-like items are available from the Texas Education Agency’s Student Assessment Division in “TAKS Information Booklets,” available at www.tea.state.tx.us/student.assessment/taks/booklets/index.html. Beginning in summer 2003 released TAKS tests will be available at www.tea.state.tx.us/index.html. Advanced Mathematics Educational Support 5 Transition to College Since 1965 enrollment in public and private higher education institutions in Texas has increased by 650,000 students. However, by 2015, Texas will need to add 500,000 more higher education students to reflect national trends and estimated needs when compared to other states.7 Thus, the level of preparation of incoming students is much more variable than was true when smaller numbers of students went on to college. Some incoming college freshmen demonstrate their academic proficiency in key subject areas by performing at acceptable levels on the state-required Texas Higher Education Assessment (THEA)8 test or on an approved alternative test. Others satisfy the college entry requirement with an acceptable performance on the exit-level TAAS or TAKS exam or on the ACT or SAT.9 In addition, Texas high school students who finish in the top 10 percent of their graduating class are guaranteed admission to state colleges and universities, regardless of the course of study they completed in high school or of their performance on THEA, ACT, or SAT. Texas institutions of higher education are challenged to make appropriate placement decisions for their incoming students. In many cases, higher education institutions have addressed any gaps in student mathematical preparation by introducing noncredit developmental courses, which too often require an inordinate amount of college mathematics department resources. Many stakeholders, including state legislators, have expressed concern about the growth of higher education developmental programs, including developmental mathematics courses, in Texas public colleges and universities. The expansion of such programs serves to illustrate the acute need for better alignment between high school mathematics programs and first-year undergraduate mathematics courses. 7 Closing the Gaps: The Texas Higher Education Plan, Texas Higher Education Coordinating Board, 2001, www.thecb.state.tx.us. 8 The THEA (formerly TASP) is an assessment program designed to ensure that students attending public institutions of higher learning or educator preparation programs in Texas have the academic skills necessary to perform effectively in college-level work. The THEA includes testing components designed to provide information about the reading, mathematics, and writing skills of students entering higher education. Students whose THEA performance shows they are not yet proficient in an academic area are required by their college or university to participate in developmental education activities For more information, see the Texas Higher Education Coordinating Board’s THEA website, www.thea.nesinc.com/. 9 For more information go to www.thecb.state.tx.us/CBRules. 6 Advanced Mathematics Educational Support MATHEMATICS EDUCATION FROM THE HIGH SCHOOL PERSPECTIVE This section addresses issues including teacher supply and retention, professional development and resources, and vertical teams. Each section describes an issue and includes general recommendations and suggestions from the AMES advisory team to higher education faculty wishing to address that issue. These are only a few of the actions a faculty member might take, and these actions would be dependent on the local situation and district, campus, teacher, and student needs. Teacher Supply and Retention The Situation Perhaps the biggest issue facing Texas high schools, particularly urban schools, is finding and retaining qualified and well-prepared mathematics teachers. Data collected by the Charles A. Dana Center since 1997 indicate that students who are taught by instructors certified in their field perform significantly better on statewide assessments; however, the same data revealed that only 78.6% of high school Algebra I courses statewide were taught by certified teachers. This percentage dropped to 68.1% in urban high schools. Recent research indicates that the shortage of teachers certified to teach Algebra I and other high school mathematics courses in Texas and in the nation will only grow worse.10 An increase in the state’s student population, combined with an increase in the number of teachers reaching retirement age and an already high teacher attrition rate, will leave Texas facing an even more critical shortage in future years. Several school districts in the state already offer signing bonuses to certified high school mathematics teachers who commit to teaching for a specified period of time. Dissatisfaction with the job is given as the main reason that teachers leave the profession.11 Recommendations • Build partnerships among your institution of higher education’s mathematics department, the college of education, and local school districts to design mathematics teacher preparation programs that help prospective teachers make the connection between the mathematics they learn at the university and the mathematics they will teach in their classrooms. • Encourage strong mathematics students who might have a talent for teaching to explore teaching as a profession. Both mathematics majors and minors should be encouraged to teach at the high school, middle school, and elementary levels. In this document’s Promising Practices section, see “UTeach” for an example of how this might be implemented. • Use the expertise of local high school master classroom teachers to help deliver components of the higher education teacher preparation program in order to provide a consistent and clear public school context to the teacher preparation program. 10 See, for example, Henke, R.R. & Zahn, L. (2001); State Board for Educator Certification (2002); Texas Education Agency (1995); and Grissmer, D. & Kirby, S. (1987). 11 Ingersoll, R. M. (1999). Advanced Mathematics Educational Support 7 • Become familiar with the financial aid available to prospective teachers through your institution as well through the Teach for Texas Conditional Grant Program and the Teach for Texas Alternative Certification Grant Program.12 • Create student organizations for cohorts of prospective mathematics teachers to provide support and collegiality among these students. Teacher Professional Development and Resources The Situation State policy requires that teachers certified since 1999 must participate in 150 hours of continuing education over five years.13 In addition, all districts require their teachers to participate in professional development on a yearly basis. To strengthen and support teachers’ content knowledge and instructional skills, K–12 schools provide teachers with professional development opportunities and resources. Ongoing, meaningful professional development is key to providing the support necessary for teachers to become more confident and successful in teaching. Too often, however, professional development opportunities are not closely targeted to the needs of the teachers or aligned with classroom instruction; consequently, the professional development has little positive effect on improving student performance. Further, classroom resources, including textbooks, may be poorly aligned to instruction. Even if the professional development or resource does meet teacher needs, teachers frequently receive little or no support in implementing either in their classrooms.14 Effective professional support should help teachers increase content knowledge and improve pedagogical skills.15 Texas high school teachers have a variety of professional development training opportunities available, offered from a variety of sources, including school districts, education service centers,16 the state, and universities. Perhaps the most comprehensive and widely available professional development in Texas is TEXTEAMS (Texas Teachers Empowered for Achievement in Mathematics and Science). Developed and managed by the Charles A. Dana Center with funding through the Texas Education Agency, TEXTEAMS provides extended learning experiences in mathematics for teachers of all levels from prekindergarten through Precalculus.17 TEXTEAMS institutes are correlated closely to the state mathematics and science Texas Essential Knowledge and Skills and are delivered by skilled mathematics and science educators who have received leadership training in the institutes. Several TEXTEAMS institutes require higher education faculty as facilitators. TEXTEAMS mathematics institutes provide teachers with mathematics instruction to deepen their understanding of content, as well as activities that can be used with their students. In addition, several universities around the state, through Eisenhower grant funding coordinated by the 12 Details are provided at www.collegefortexans.com. Texas Administrative Code, Title 19, Part 7, Chapter 232, Subchapter R, Rule §232.850. This rule can be found at www.sos.state.tx.us/tac/index.html. 14 Loucks-Horsley, Hewson, Love, and Stiles, 1998; Sowder, Philipp, Armstrong, and Schappelle, 1998. 15 Ball and Cohen, 1999; Briars, 2000; U.S. DOE, 2000; Kennedy, 1999. 16 Texas is divided into 20 geographic regions, each having an education service center. These centers coordinate with the Texas Education Agency to provide educational services to schools within their region. 17 Since 1996 TEXTEAMS institutes have provided intensive, in-depth, content-based professional development for more than 200,000 educators. These institutes are offered by more than 700 certified leaders throughout the state. For more information, see the TEXTEAMS website at www.texteams.org. 13 8 Advanced Mathematics Educational Support Texas Higher Education Coordinating Board, offer localized programs to help improve teacher learning and student success. Local school districts and education service centers also develop and provide professional development opportunities for teachers. Recommendations • Work with local districts to determine teacher professional development needs. • Work with local districts and college of education colleagues to design, deliver, and support ongoing, intensive professional development (face-to-face, online, and distance learning) that meets teacher needs. • Become TEXTEAMS university leaders and use TEXTEAMS materials with preservice and inservice teachers and graduate students. • Design college-credit summer courses for high school mathematics teachers that align with teacher needs. Become familiar with grant funds available to college faculty for this purpose through the Teacher Quality Higher Education Grants Program.18 • Investigate the opportunity for your institution to offer College Board Advanced Placement Summer Institutes that are designed to provide professional development for teachers of designated AP courses.19 • Become knowledgeable about high school curriculum resources, including textbooks, assessments, technology, and supplemental materials, that support teaching the content described in the TEKS. Offer to assist in resource development and evaluation when appropriate. • Work with teachers to develop rich lessons that address multiple interrelated concepts instead of surface lessons that address single concepts in isolation from each other. • Use existing master’s degree programs to attract teachers who need to learn or relearn mathematics content. • Serve as a mentor or resource person for teachers and students. This might include working with schools to build student support structures that directly improve student achievement. • Make presentations on mathematics content and topics relevant to high school mathematics teachers at state, local, and national professional conferences. For example, the Conference for the Advancement of Mathematics Teaching (CAMT) and regional and national conferences of the National Council of Teachers of Mathematics (NCTM) encourage session proposals from university faculty.20 18 Details are provided on the Texas Higher Education Coordinating Board’s website at www.thecb.state.tx.us. For information regarding how to become involved with AP Summer Institutes, contact the Southwestern Regional Office of the College Board in Austin, Texas, at 512-891-8400 or 800-999-9139. 20 Details are provided at www.tenet.edu/camt and www.nctm.org. 19 Advanced Mathematics Educational Support 9 Vertical Teaming The Situation In an effort to strengthen their mathematics programs, many districts have adopted the strategy of using vertical teams to achieve a well-aligned, seamless mathematics curriculum. Such a curriculum can be described as one that presents a thoughtful progression of concepts within courses and from one course to another, avoiding redundancies, gaps, and jarring transitions. Vertical team denotes a team made up of teachers in a given content area from a sequence of grade levels who work together to build a strong instructional program, with a common goal of ensuring success for all students. The College Board promotes vertical teaming as an Advanced Placement strategy for teachers in a specific content area in grades 6–12 to foster the readiness of junior- and senior-level high school students for the AP tests given for the purpose of accruing college credits. In recent years, the concept has expanded to teachers in grades K–12 and has shown potential for K–16. Many districts have implemented “vertical teams” that include only AP and Pre-AP teachers and do not share a common goal of ensuring success for all students, or even address mathematics programmatic issues in substantive ways. Unfortunately, many of these teams become dysfunctional and disband after a year or two. In an unpublished pilot study conducted by the Dana Center between 1999 and 2001 the most effective vertical teams included teachers from every level of the subject area—not just the teachers of Pre-AP or AP students—who focused on programmatic issues and held high expectations for all students. Recommendations • Become familiar with various resources designed to build and support vertical teams, especially the Advanced Placement Program Mathematics Vertical Teams Toolkit (1998).21 • If a local school district has implemented vertical teaming: Learn about the district’s implementation experience and work with district leadership to identify potential areas of concern. Become part of the district’s vertical team and work to strengthen the district’s implementation of vertical teaming (see Appendix C). 21 The Dana Center, in conjunction with the College Board, has developed the Advanced Placement Program Mathematics Vertical Teams Toolkit, a resource guide for implementing the vertical teaming process to support strong mathematics instruction. (This resource is available from the Dana Center and from the College Board.) Chapter 2 of the Toolkit proposes activities for initial team meetings that will help team members build trust with one another, begin to explore curriculum issues, and set preliminary goals for the team. It is critical to the success of the team to spend adequate time building trust among team members. 10 Advanced Mathematics Educational Support • If a local school district has not implemented vertical teaming: Explore with district leadership the potential benefits of establishing a vertical team. Share information with the district leadership about the available resources supporting vertical teams. Help the district form a K–16 vertical team (see Appendix C). Advanced Mathematics Educational Support 11 MATHEMATICS EDUCATION FROM THE HIGHER EDUCATION PERSPECTIVE This section addresses issues of concurrent enrollment, standards for high school calculus and statistics, tenure and promotion, undergraduate teacher preparation, and alternative teacher certification. Each section briefly describes an issue and includes general recommendations from the AMES advisory team. Concurrent Enrollment The Situation The practice of having high school students enroll in college courses while still in high school is known as concurrent or dual enrollment. Although these terms are often used interchangeably, in the TEKS the term concurrent enrollment is used to describe the practice in which students receive high school and college credit for the same course.22 Local implementation of concurrent enrollment practices varies widely. For example, some students may wish to take an advanced course that is not offered at their school, so they go to a nearby college to take the course. In other cases, students may take the college course on their high school campus. While concurrent enrollment appears to offer students enhanced educational opportunities, in practice issues can arise. Identifying these issues can serve as a starting place for discussion among high school educators, higher education faculty, parents, and other stakeholders. One issue is that not all concurrent enrollment scenarios offer safeguards guaranteeing that the course experience is truly at the collegiate level. Contributing to this problem are inconsistencies in course content standards and faculty expertise. For example, if this course is taught by a high school teacher without collaboration with university or college faculty, the content may not be consistent with that of the same course taught at the college or university level. There may also be a difference in the depth of mathematical content knowledge between high school and higher education faculty. Another issue is that the college-level course content that the student studies for concurrent enrollment credit may not align with the content in the high school course for which the student is given high school mathematics credit. These high school courses include Algebra II, Precalculus, AP Statistics, AP Calculus AB, and AP Calculus BC. The content for these high school courses is described by state law in the Texas Essential Knowledge and Skills (see Appendix A). The TEKS also specify requirements for concurrent enrollment credit: Section 111.60. Concurrent Enrollment in College Courses. (a) General requirements. Students shall be awarded one-half credit for each semester of successful completion of a college course in which the student is concurrently enrolled while in high school. (b) Content requirements. In order for students to receive state graduation credit for concurrent enrollment courses, content requirements must meet or exceed the essential knowledge and skills in a given course. For example, if students complete College Algebra and receive concurrent enrollment credit for Algebra II, the College Algebra course content must address all the TEKS in Algebra II. 22 The term dual enrollment would then be used to describe the practice in which high school students enroll in college courses, but the credit for those college courses does not count toward high school graduation requirements. 12 Advanced Mathematics Educational Support Recommendations • Engage in dialogue with public school personnel about critical issues regarding concurrent enrollment, such as course content, alignment with other courses, and faculty expertise. • Collaborate with high school mathematics teachers to mutually ensure the rigor of courses offered for concurrent credit. Build collegial relationships with these teachers by building a network, meeting together on a regular basis, and attending professional development and conferences together. • Work with public school personnel to develop and implement appropriate standards and policies for concurrent enrollment. • Ensure that concurrent enrollment instructors are active members of both the high school and the university or college mathematics departments. It is essential that higher education faculty on a high school campus be engaged with high school mathematics faculty. For example, a faculty member teaching College Algebra to students receiving concurrent credit should communicate with teachers of pertinent courses on the high school campus. • Collaborate with public school personnel to ensure that concurrent enrollment courses meet the requirements described in the Texas Essential Knowledge and Skills before awarding state graduation credit. Standards for High School Calculus and Statistics The Situation Any student who takes calculus in secondary school and performs satisfactorily in that course should place out of the comparable college calculus course. Calculus courses that meet the level of rigor necessary for this to occur must be preceded by four full years of mathematical preparation in algebra, geometry, trigonometry, analytic geometry, and elementary functions (College Board, 2002). Students who enroll in a high school statistics course should have successfully completed three full years of mathematical preparation in algebra and geometry, including successful completion of a rigorous Algebra II course. As a response to this recommendation, Texas designated AP Calculus AB and AP Calculus BC as the only high school calculus courses approved for graduation credit. Similarly, AP Statistics is the only statistics course approved for graduation credit. However, contrary to the state’s intent, some schools offer a non-AP calculus or statistics course and give credit for Independent Study or award a local credit. According to the AMES advisory team, many of their university colleagues do not support the AP program because of their negative experiences with many students who took Advanced Placement courses in high school. For example, many students claim, “I had AP calculus in high school,” yet these students did not take the AP exam and frequently can demonstrate only some rudimentary calculus techniques. It is very probable that such students did not successfully complete a true AP course; the standards for AP courses are extremely high. For example, an AP calculus course is built around a course description reflective of a college calculus course. Advanced Mathematics Educational Support 13 There are actually two AP calculus courses, entitled Calculus AB and Calculus BC. (For a list of topics for these courses, see Appendix B.) The AB course and exam cover roughly the first college semester of calculus, through integration by substitution and separation of variables; the BC course and exam cover the first two semesters of calculus, including numeric and Taylor series, and the calculus of parametric, polar, and vector functions of one variable. Like AP Calculus, AP Statistics is built around a course description and examination designed to reflect the content of a typical introductory college course in statistics. The purpose of the AP course in statistics is to introduce students to major concepts and tools for collecting, analyzing, and drawing conclusions from data. The AP course is organized around four broad topics: data exploration, experimental design, modeling, and inference. This course provides students with an experience equivalent to that of a one-semester, introductory non-calculusbased college course in statistics. (For a list of course topics, see Appendix B.) All three AP mathematics courses are designed to culminate in students taking an AP exam in May. These exams are the same throughout the country and are graded by a select gathering of highly qualified college and high school calculus or statistics teachers. The existence of these independently administered and graded exams, together with the high standards of the grading process, safeguard the quality of the program. Recommendations 23 24 • Become knowledgeable about the public school curriculum to help teachers maintain high standards in all mathematics courses so that all students are adequately prepared for their next mathematics course. (See Appendix A for the high school Texas Essential Knowledge and Skills for Mathematics.) • Collaborate with local school districts to educate high school seniors, parents, and counselors regarding the importance of mathematics in students’ options for a college major. Work with local school districts to implement policies that ensure that students take a rigorous mathematics course their senior year. • Work with local school districts to implement policies that ensure that local high schools that offer calculus and statistics courses offer only AP Calculus and/or AP Statistics. • Encourage school districts to develop a policy requiring every student enrolled in an AP course to take the corresponding AP exam. • Become an easily accessible resource for local AP teachers. Support the development of AP teachers by becoming involved in College Board AP summer institutes and providing ongoing professional development and support.23 Encourage school districts to send their teachers to AP training. • Become a reader for the AP exams.24 For information about how to become involved with AP summer institutes, go to www.collegeboard.org. For information on how to become a reader for the AP exams, go to www.collegeboard.org. 14 Advanced Mathematics Educational Support Tenure and Promotion The Situation Time constraints are a barrier to effective collaboration between higher education and public schools. For many college faculty, time spent on research is crucial, because tenure, promotion, and merit raises rely heavily on research productivity. Although this problem is not easily solved, major research-funding organizations, such as the National Science Foundation, require K–12 educational outreach components for many of the science and mathematics research grants awarded. A university culture that supports and welcomes coordination between mathematics faculty and education faculty and that desires strong ties with K–12 educators will make it easier for educators with an interest in bridging the gap between secondary and college instruction to act on their inclinations and develop strong connections that help both the secondary classroom teacher and the university professoriate. Several universities who see teacher inservice and preservice preparation as an important component of their mission have begun to value faculty work with K–12 educators for tenure and promotion. For example, in this document’s Promising Practices section, see “Systemic/Programmatic Administration Support.” Recommendations • Work to establish policies in your department and college that ensure that meaningful involvement with K–12 educators is valued toward tenure. (In this document’s Promising Practices section, see “Systemic/Programmatic Administration Support.”) • Become an active participant in a statewide network that is considering issues around revision of tenure policies. For example, the annual October Preservice Conference25 provides a forum for discussion of these issues. • Build meaningful collaborations with public school and college of education counterparts that will support teachers and lead to student success. If a measure of student success can be attributed to mathematics faculty involvement, this strengthens the argument for meaningful involvement in high school counting toward tenure. (In this document’s Promising Practices section, see “Systemic/Programmatic Administration Support.”) • Advance the knowledge base for improving student performance in mathematics by valuing and rewarding faculty research on secondary mathematics instructional practices. 25 The Charles A. Dana Center has hosted the annual October Preservice Conference for mathematics, science, and education faculty for the past eight years, with participation of faculty from across the state. For more information, see www.utdanacenter.org. Advanced Mathematics Educational Support 15 Undergraduate Teacher Preparation The Situation A number of reports (Kilpatrick, Swafford, and Findell, 2001; Ma, 1999; U.S. DOE, 2002) underscore the importance of strong content knowledge for all mathematics teachers and specifically emphasize the need for a focus on the mathematics that they will be expected to teach. State and national mandates urge greater involvement by more university mathematicians in the mathematical preparation of teachers.26 The Texas State Board for Educator Certification in April 2000 responded to these challenges by adopting new certification standards for Texas teachers in each of the major subject areas. These new certification standards move higher education from a credit-based to a standards-based system for teacher preparation, and include a high-stakes accountability system for colleges and universities that prepare teachers. These new certification standards delineate, for the first time, not how many courses beginning educators have to take, but rather what knowledge and skills they must acquire. These standards also specify content knowledge and teaching skills for teachers of specific grade levels—that is, early childhood–grade 4, grades 4–8, and grades 8–12. These teacher certification standards are connected to the TEKS, the state’s curriculum guidelines for what public school students must know and be able to do. New teacher certification examinations—the Texas Examinations of Educator Standards (TExES)—are aligned with the new standards. An institution that prepares teachers is accredited based on how well its prospective teachers perform on this test.27 This standards-based program allows for greater flexibility and creativity in the design of teacher preparation programs, while ensuring greater uniformity and quality in teacher preparation. It is also forcing a reexamination of the mathematics preparation of teachers across the state and creating a need for greater collaboration between colleges of education and mathematics departments in designing high-quality programs that meet state and national standards.28 Recommendations • Department of mathematics faculty should examine current curriculum and programs in relationship to the SBEC beginning teacher certification standards and ensure that the mathematics content required for certification is infused within the required mathematics courses. 26 For example, the NSF mathematics and science partnership grants require substantial involvement of university mathematics faculty in improving K–16 education. In addition numerous reports recommend the deep involvement of mathematics faculty, both with the preparation of prospective teachers and with K–12 schools (American Mathematical Society, 2001; U.S. DOE, 2000). 27 For additional information, see the State Board for Educator Certification website, www.sbec.state.tx.us. 28 A project from the Charles A. Dana Center currently underway, “Strengthening and Supporting Standards-Based Mathematics Teacher Preparation (S3MTP),” will provide opportunities for faculty collaboration, professional recognition, and leadership, and develop resources for implementing standards-based teacher preparation and certification. Details are provided at www.utdanacenter.org/mathematics/highered/projects.html. 16 Advanced Mathematics Educational Support • Network with mathematics faculty on your campus and at other institutions who are involved with teacher education to build a solid mathematics program to prepare teachers. • Support faculty members in your institution who choose to invest their time and apply their expertise to the preparation of mathematics teachers. Alternative Teacher Certification The Situation In an effort to alleviate the severe shortage of mathematics teachers in Texas, a variety of alternative certification programs have been developed and offered by education service centers, school districts, colleges and universities, and other entities. Alternative certification is available to people who wish to become teachers and who hold a baccalaureate degree but do not have a teaching certificate. Because of the No Child Left Behind Act of 2001, all alternative certification programs will need to be reviewed to meet the requirements for producing highly qualified teachers. Recommendations • Become informed about and involved in the alternative certification programs in your region.29 • Become familiar with the financial aid available to alternative certification candidates through your institution as well as through the Teach for Texas Alternative Certification Grant Program.30 29 For more information regarding alternative certification, go to the State Board for Educator Certification website at www.sbec.state.tx.us. 30 Details are provided at www.collegefortexans.com. Advanced Mathematics Educational Support 17 PROMISING PRACTICES This section includes samples of model programs and promising practices that address selected issues from the previous two sections. It includes brief descriptions of the programs with weblinks that offer additional information. This section is not intended to include all programs available, nor are we suggesting that programs can be easily transported from one institution to another. However, it does offer a snapshot of responses to concerns about teacher preparation and teacher quality that can be used to spur discussion and generate ideas that can lead to improved teacher preparedness and connections between high schools and universities. Master Teacher Summer Institutes The University of Texas at Austin www.mtsi.utexas.org The Master Teacher Summer Institutes at The University of Texas at Austin are designed for teachers of Advanced Placement courses to work with university content faculty and master teachers. The goal of these month-long professional development experiences is to increase the diversity and academic skills of Texas public school students enrolled in pre-AP and AP courses and to improve the content, technological, and content-specific pedagogical knowledge of the teacher participants. Through these institutes, high school teachers are able to work with university faculty in a way that increases teacher knowledge of the content they teach, offers suggestions for how to use that new knowledge in the classroom, and helps them build their own program of professional development that will lead to continued improvement of their teaching. The Master Teacher Summer Institutes are offered through the College of Natural Sciences and the College of Liberal Arts. AP Summer Institute Baylor University Since 1995 Baylor University has hosted an annual summer Advanced Placement Institute for high school teachers. This AP Institute provides an excellent opportunity for teachers and professors to learn from each other about contemporary students, curriculum, and academic expectations. The high school teachers benefit from having access to university professors who are experts in their fields, to relevant research and tools in the university library, and to special events designed specifically for them. Baylor benefits from the contact with these teachers of the academically gifted young people who are among its prospective students. To get the institute started, Baylor first contacted the Southwestern Regional Office of the College Board in Austin. This office provided not only guidance on the format for the institute, but also a list of potential consultants to conduct the workshops. The College Board also helped with publicity for the institute, provided resource materials and certificates of attendance for the participants, and shared information about state reimbursements for the cost of the institute to qualified participants. 18 Advanced Mathematics Educational Support The Rice University School Mathematics Project rusmp.rice.edu The Rice University School Mathematics Project was created in 1987 with funding from a three-year National Science Foundation grant. It was initially designed to include 48 middle and high school teachers from a diverse group of local schools and school districts. In the first year these teachers participated in an intensive six-week summer instructional program on the Rice University campus. The goals of the program from its onset were: • active involvement of research mathematicians, mathematics educators, and classroom teachers in the development of the mathematical content taught in the program, • collaboration of mathematics educators and master teachers to ensure that models of content and instructional delivery were appropriate for the secondary classroom, • use of an inquiry and problemsolving approach whenever possible, • development of participants to serve as “lead teachers” at their respective campuses so that the training could be disseminated more broadly, • development of content-based curricular materials by participants, • appropriate use of technology, and • interaction between secondary teachers and research mathematicians. The Rice University School Mathematics Project continued with funding from private and local foundations. It includes creating academic-year courses for teachers in mathematics content and pedagogy, planning and writing curriculum and supplementary materials for local school districts, and developing innovative ways to bridge the gap between teacher training and classroom implementation. One of these programs, the RUSMP Urban Program, combines teacher training with summer school and academic year instruction and has been nationally acclaimed as a leading staff development model. The project provides training through its various programs to about 600 teachers annually, including 120 each year in the Summer Campus Program, which is now in its 17th year. As the project has expanded, it has evolved from its initial beginnings as part of the Mathematics Department into a center within the School of Natural Sciences. Information Technology in Science Center for Teaching and Learning Texas A&M University its.tamu.edu The Information Technology in Science Center for Teaching and Learning, a partnership between the College of Science and the College of Education at Texas A&M University, is an interdisciplinary graduate program that seeks to replenish the nation’s supply of education specialists in science, mathematics, and technology through learner-centered opportunities Advanced Mathematics Educational Support 19 involving scientists, mathematicians, education researchers. and education practitioners. The Center’s three main goals are (1) production of education specialists, (2) creation of new knowledge through research on the impact of information technology in the form of modeling, visualization, and interaction with complex data sets on learning and teaching science and mathematics, and (3) development and dissemination of quality professional development experiences structured around the impact of information technology on learning and teaching science and mathematics in grades 7–12. The ITS Center’s innovative approach combines faculty, graduate students, and master teachers who work together as a science-based team to develop key questions, design and conduct research, and provide professional development experiences while gaining valuable technological skills. Participants are part of a research program based on integration, coordination, and application. In addition to integrated science, education, and technology experiences, participants are involved in research experiences coordinated by the center to ensure that each participant’s individual project becomes part of a coherent whole. The ITS program offers direct applications and connections to classrooms across the state and nation by transferring current science, mathematics, and engineering research into grades 7–12 classrooms to assist teachers in integrating technology into the science curriculum. UTeach The University of Texas at Austin www.uteach.utexas.edu The UTeach program at The University of Texas at Austin is a teacher-preparation program at a traditional four-year institution. UTeach stands as a model because of the intensity with which the Colleges of Natural Science and Education and the Austin Independent School District have collaborated to create a teacher-preparation program that is attractive to students and that prepares them to teach with knowledge of research-based practices. UTeach was developed in consultation with a group of master high school teachers and the Texas State Board for Educator Certification, according to the new guidelines for teacher certification and new national and state standards for K–12 education in mathematics and science. All UTeach degree plans can be completed in four years. Teaching techniques, field experience, the study of mathematics and science, and certification are fully integrated. The courses were codeveloped for this program by the College of Natural Sciences and the College of Education. Hallmarks of the UTeach program include:31 • 31 Active recruitment and support of natural science undergraduates who are interested in careers in secondary math and science education. Support includes tuition reimbursement, placement with small cohorts of students, paid internships, and guidance by master teachers. Material that follows is from the UTeach website (www.uteach.utexas.edu). 20 Advanced Mathematics Educational Support • Emphasis on preparing teachers who will be knowledgeable of their discipline, experienced with involving students in scientific inquiry, and practiced in employing new technologies to enhance student learning. • A revised, streamlined professional education sequence drawing on research on learning, standards-based curricula, multiple forms of assessment, and proven strategies for achieving equity and integrating technology into math and science education. • Integrated preservice and content experiences which prepare UTeach students to teach all levels of material—from the core curriculum to Advanced Placement courses—to students of diverse cultural and socioeconomic backgrounds. • Program flexibility with multiple entry points (from freshman to postbaccalaureate), integrated degree plans, and proficiency-based assessment, including the development of individual teaching portfolios. In addition to natural sciences and education faculty, UTeach employs master teachers in the program as instructors, advisers, and field supervisors. Since it began in 1997, UTeach has grown to over 250 students, and is already graduating over 50 teachers per year. Student retention as undergraduates is much higher than College of Natural Sciences averages, and nearly 40% of the first teaching graduates have been minority students. Systemic/Programmatic Administration Support Stephen F. Austin State University www.sfasu.edu/math One major obstacle university mathematics faculty face when becoming involved in public school education is the criteria for merit and tenure at the university level. (See the section “Tenure and Promotion” in this document’s section on Mathematics Education from the Higher Education Perspective.) In the Stephen F. Austin State University Department of Mathematics and Statistics, active engagement with K–12 schools by mathematics faculty is counted toward merit and tenure, as is professional development or service. Department policy supported by the university states that funded grants, including those related to public school teaching, are counted the same as a refereed publication in a professional journal. With a funded grant project, consideration is given to the number of people affected by the project, the money that is brought to the university, and the publicity and goodwill associated with the grant project. Efforts of the Department of Mathematics and Statistics at SFASU to improve teacher preparation and quality receive the full support of the upper administration (deans, vice presidents, president, and governing board). The mathematics and statistics department chairman actively promotes the idea that the university’s upper administration will support any efforts that improve the success rate in mathematics. Most upper administrators are interested in the university’s students being successful in their chosen program, which in many instances includes a significant amount of mathematics. In part because of this work to improve K–12 education, students who enter the university will be better prepared. Thus, retention rates will rise. Advanced Mathematics Educational Support 21 As of 2002 SFASU had seven graduate mathematics courses and six undergraduate courses designed specifically to meet the needs of preservice and inservice teachers. One incentive at SFASU to encourage faculty members to become involved in mathematics education is release time given for faculty to develop new mathematics education courses. Even at SFASU, with its favorable merit and tenure policies, attracting mathematics faculty members who become involved in the preparation and/or professional development of K–12 teachers is difficult. Encouraging faculty after they come to the university is the easy part, and it helps to have an administration that supports such faculty members’ efforts. Lamar State College–Orange Mathematics Institutes Lamar State College–Orange, Texas www.orange.lamar.edu/Academics/Academic_Eisenhower.htm Providing high-quality professional development to PK–12 teachers is a challenge. For those college or university personnel with an Eisenhower Higher Education Grant32 project, a special challenge has been the length of the training that this grant requires be provided, which is approximately 100 hours. Since reviewers from throughout the country judge Eisenhower proposals, it is also important that the content provided in training funded by Eisenhower grants be of high quality. Beginning in 1999, the mathematical content of the Lamar State College–Orange Eisenhowerfunded mathematics institutes was built around TEXTEAMS (Texas Teachers Empowered for Achievement in Mathematics and Science) institutes. (Refer to “Teacher Professional Development and Resources” in the Mathematics Education from the High School Perspective section of this document.) TEXTEAMS materials are designed to help educators understand and implement the mathematics and science Texas Essential Knowledge and Skills (TEKS) and to support student success on TEKS-based assessments, such as the Texas Assessment of Knowledge and Skills (TAKS). Each TEXTEAMS institute is developed around unifying themes and concepts articulated in the TEKS. Additionally, TEXTEAMS institutes focus on helping educators think deeply about the “big ideas” found in the mathematics and science curriculum, such as proportionality in grades 6–8. The TEXTEAMS institutes provide the foundation upon which teachers can discuss, explore, and experience mathematical content in a hands-on environment, using appropriate technology. Participating teachers are given time to thoroughly explore the mathematics content. For example, the TEXTEAMS institute Algebra I: 2000 and Beyond was designed as five-day, 30hour professional development. In the Lamar State College–Orange Mathematics Institute, teachers spend 40 hours during the summer and 21 hours in the fall semester exploring the content of Algebra I. The academic year provides LSCO institute participants the opportunity to implement with their students the content learned in the summer institutes. After teaching the content in their classrooms, LSCO institute participants have the opportunity to revisit the 32 Since 1985–86 the Texas Higher Education Coordinating Board has provided grants through the federal Eisenhower Professional Development Grants Program in mathematics and science for K–12 school personnel. Beginning with the 2003–04 academic year, the Texas Higher Education Coordinating Board is funding the Teacher Quality Professional Development Grants Program to replace Eisenhower grants. Grants are awarded to higher education institutions in April for implementation from May 1 of the funding start year to September 30 of the following year. 22 Advanced Mathematics Educational Support institute experiences in a structured setting and discuss successes, problems, and/or suggested modifications to their classroom instruction. This same principle has been applied to other LSCO institutes targeted at both middle school and high school programs. One major advantage of using TEXTEAMS materials as the foundation for LSCO Eisenhower institutes is the hours of work saved by institute personnel in the development of content for the courses. The advisory teams, authors, and presenters of the TEXTEAMS institutes have already completed much work when the institutes are ready for the training sessions. The higher education instructors have the opportunity to build upon their work rather than developing content from scratch. Other advantages of using TEXTEAMS materials within the LSCO mathematics institutes include: • TEKS-based activities are used that are already teacher-tested and student-tested. • Some of the institutes include sample student work, which further enhances the content. • The institutes provide an opportunity for higher education personnel to observe teachers discussing the content, how it will work with their students, and other issues related to implementation in the classroom. For higher education faculty who have never received or conducted an Eisenhower-funded professional development project, the use of the TEXTEAMS materials provides a blueprint to begin their involvement with the professional development for PK–12 school teachers. The higher education faculty can supplement or modify the content to whatever extent they choose. Advanced Mathematics Educational Support 23 APPENDIX A CHAPTER 111. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS FOR MATHEMATICS Subchapter C. High School33 Statutory Authority: The provisions of this Subchapter C issued under the Texas Education Code, §28.002, unless otherwise noted. §111.31. Implementation of Texas Essential Knowledge and Skills for Mathematics, Grades 9-12. The provisions of this subchapter shall be implemented beginning September 1, 1998, and at that time, shall supersede §75.63(e)-(g) of this title (relating to Mathematics). Source: The provisions of this §111.31 adopted to be effective September 1, 1996, 21 TexReg 7371. §111.32. Algebra I (One Credit). (a) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students will continue to build on this foundation as they expand their understanding through other mathematical experiences. (2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students use symbols in a variety of ways to study relationships among quantities. (3) Function concepts. Functions represent the systematic dependence of one quantity on another. Students use functions to represent and model problem situations and to analyze and interpret relationships. (4) Relationship between equations and functions. Equations arise as a way of asking and answering questions involving functional relationships. Students work in many situations to set up equations and use a variety of methods to solve these equations. (5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof. 33 The high school mathematics TEKS can be found on the Texas Education Agency website at http://www.tea.state.tx.us/rules/tac/chapter111/ch111c.html. 24 Advanced Mathematics Educational Support (b) Foundations for functions: knowledge and skills and performance descriptions. (1) (2) (3) The student understands that a function represents a dependence of one quantity on another and can be described in a variety of ways. The student uses the properties and attributes of functions. The student understands how algebra can be used to express generalizations and recognizes and uses the power of symbols to represent situations. Advanced Mathematics Educational Support Following are performance descriptions. (A) The student describes independent and dependent quantities in functional relationships. (B) The student gathers and records data, or uses data sets, to determine functional (systematic) relationships between quantities. (C) The student describes functional relationships for given problem situations and writes equations or inequalities to answer questions arising from the situations. (D) The student represents relationships among quantities using concrete models, tables, graphs, diagrams, verbal descriptions, equations, and inequalities. (E) The student interprets and makes inferences from functional relationships. Following are performance descriptions. (A) The student identifies and sketches the general forms of linear (y = x) and quadratic (y = x2) parent functions. (B) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations. (C) The student interprets situations in terms of given graphs or creates situations that fit given graphs. (D) In solving problems, the student collects and organizes data, makes and interprets scatterplots, and models, predicts, and makes decisions and critical judgments. Following are performance descriptions. (A) The student uses symbols to represent unknowns and variables. (B) Given situations, the student looks for patterns and represents generalizations algebraically. 25 (4) (c) Following are performance descriptions. (A) The student finds specific function values, simplifies polynomial expressions, transforms and solves equations, and factors as necessary in problem situations. (B) The student uses the commutative, associative, and distributive properties to simplify algebraic expressions. Linear functions: knowledge and skills and performance descriptions. (1) (2) 26 The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student understands that linear functions can be represented in different ways and translates among their various representations. The student understands the meaning of the slope and intercepts of linear functions and interprets and describes the effects of changes in parameters of linear functions in real-world and mathematical situations. Following are performance descriptions. (A) The student determines whether or not given situations can be represented by linear functions. (B) The student determines the domain and range values for which linear functions make sense for given situations. (C) The student translates among and uses algebraic, tabular, graphical, or verbal descriptions of linear functions. Following are performance descriptions. (A) The student develops the concept of slope as rate of change and determines slopes from graphs, tables, and algebraic representations. (B) The student interprets the meaning of slope and intercepts in situations using data, symbolic representations, or graphs. (C) The student investigates, describes, and predicts the effects of changes in m and b on the graph of y = mx + b. (D) The student graphs and writes equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept. (E) The student determines the intercepts of linear functions from graphs, tables, and algebraic representations. (F) The student interprets and predicts the effects of changing slope and y-intercept in applied situations. Advanced Mathematics Educational Support (G) (3) (4) The student formulates equations and inequalities based on linear functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student formulates systems of linear equations from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student relates direct variation to linear functions and solves problems involving proportional change. Following are performance descriptions. (A) The student analyzes situations involving linear functions and formulates linear equations or inequalities to solve problems. (B) The student investigates methods for solving linear equations and inequalities using concrete models, graphs, and the properties of equality, selects a method, and solves the equations and inequalities. (C) For given contexts, the student interprets and determines the reasonableness of solutions to linear equations and inequalities. Following are performance descriptions. (A) The student analyzes situations and formulates systems of linear equations to solve problems. (B) The student solves systems of linear equations using concrete models, graphs, tables, and algebraic methods. (C) For given contexts, the student interprets and determines the reasonableness of solutions to systems of linear equations. (d) Quadratic and other nonlinear functions: knowledge and skills and performance descriptions. (1) The student understands that the graphs of quadratic functions are affected by the parameters of the function and can interpret and describe the effects of changes in the parameters of quadratic functions. Advanced Mathematics Educational Support Following are performance descriptions. (A) The student determines the domain and range values for which quadratic functions make sense for given situations. (B) The student investigates, describes, and predicts the effects of changes in a on the graph of y = ax2. (C) The student investigates, describes, and predicts the effects of changes in c on the graph of y = x2 + c. (D) For problem situations, the student analyzes graphs of quadratic functions and draws conclusions. 27 (2) (3) The student understands there is more than one way to solve a quadratic equation and solves them using appropriate methods. The student understands there are situations modeled by functions that are neither linear nor quadratic and models the situations. Following are performance descriptions. (A) The student solves quadratic equations using concrete models, tables, graphs, and algebraic methods. (B) The student relates the solutions of quadratic equations to the roots of their functions. Following are performance descriptions. (A) The student uses patterns to generate the laws of exponents and applies them in problem-solving situations. (B) The student analyzes data and represents situations involving inverse variation using concrete models, tables, graphs, or algebraic methods. (C) The student analyzes data and represents situations involving exponential growth and decay using concrete models, tables, graphs, or algebraic methods. Source: The provisions of this §111.32 adopted to be effective September 1, 1996, 21 TexReg 7371. 28 Advanced Mathematics Educational Support §111.33. Algebra II (One-Half to One Credit). (a) (b) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. (2) Algebraic thinking and symbolic reasoning. Symbolic reasoning plays a critical role in algebra; symbols provide powerful ways to represent mathematical situations and to express generalizations. Students study algebraic concepts and the relationships among them to better understand the structure of algebra. (3) Functions, equations, and their relationship. The study of functions, equations, and their relationship is central to all of mathematics. Students perceive functions and equations as means for analyzing and understanding a broad variety of relationships and as a useful tool for expressing generalizations. (4) Relationship between algebra and geometry. Equations and functions are algebraic tools that can be used to represent geometric curves and figures; similarly, geometric figures can illustrate algebraic relationships. Students perceive the connections between algebra and geometry and use the tools of one to help solve problems in the other. (5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities and model mathematical situations to solve meaningful problems. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof. Foundations for functions: knowledge and skills and performance descriptions. (1) The student uses properties and attributes of functions and applies functions to problem situations. Advanced Mathematics Educational Support Following are performance descriptions. (A) For a variety of situations, the student identifies the mathematical domains and ranges and determines reasonable domain and range values for given situations. (B) In solving problems, the student collects data and records results, organizes the data, makes scatterplots, fits the curves to the appropriate parent function, interprets the results, and proceeds to model, predict, and make decisions and critical judgments. 29 (2) (3) (c) The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situations. Following are performance descriptions. (A) The student uses tools including matrices, factoring, and properties of exponents to simplify expressions and transform and solve equations. (B) The student uses complex numbers to describe the solutions of quadratic equations. (C) The student connects the function notation of y = and f(x) =. Following are performance descriptions. (A) The student analyzes situations and formulates systems of equations or inequalities in two or more unknowns to solve problems. (B) The student uses algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities. (C) For given contexts, the student interprets and determines the reasonableness of solutions to systems of equations or inequalities. Algebra and geometry: knowledge and skills and performance descriptions. (1) (2) 30 The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student connects algebraic and geometric representations of functions. The student knows the relationship between the geometric and algebraic descriptions of conic sections. Following are performance descriptions. (A) The student identifies and sketches graphs of parent functions, including linear (y = x), quadratic (y = x2), square root (y = √x), inverse (y = 1/x), exponential (y = ax), and logarithmic (y = loga x) functions. (B) The student extends parent functions with parameters such as m in y = mx and describes parameter changes on the graph of parent functions. (C) The student recognizes inverse relationships between various functions. Following are performance descriptions. (A) The student describes a conic section as the intersection of a plane and a cone. Advanced Mathematics Educational Support . (d) (B) In order to sketch graphs of conic sections, the student relates simple parameter changes in the equation to corresponding changes in the graph. (C) The student identifies symmetries from graphs of conic sections. (D) The student identifies the conic section from a given equation. (E) The student uses the method of completing the square. Quadratic and square root functions: knowledge and skills and performance descriptions. (1) (2) (3) The student understands that quadratic functions can be represented in different ways and translates among their various representations. The student interprets and describes the effects of changes in the parameters of quadratic functions in applied and mathematical situations. The student formulates equations and inequalities based on quadratic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Advanced Mathematics Educational Support Following are performance descriptions. (A) For given contexts, the student determines the reasonable domain and range values of quadratic functions, as well as interprets and determines the reasonableness of solutions to quadratic equations and inequalities. (B) The student relates representations of quadratic functions, such as algebraic, tabular, graphical, and verbal descriptions. (C) The student determines a quadratic function from its roots or a graph. Following are performance descriptions (A) The student uses characteristics of the quadratic parent function to sketch the related graphs and connects between the y = ax2 + bx + c and the y = a(x - h)2 + k symbolic representations of quadratic functions. (B) The student uses the parent function to investigate, describe, and predict the effects of changes in a, h, and k on the graphs of y = a(x - h)2 + k form of a function in applied and purely mathematical situations. Following are performance descriptions. (A) The student analyzes situations involving quadratic functions and formulates quadratic equations or inequalities to solve problems. (B) The student analyzes and interprets the solutions of quadratic equations using discriminants and solves quadratic equations using the quadratic formula. 31 (4) (e) The student formulates equations and inequalities based on square root functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student compares and translates between algebraic and graphical solutions of quadratic equations. (D) The student solves quadratic equations and inequalities. Following are performance descriptions. (A) The student uses the parent function to investigate, describe, and predict the effects of parameter changes on the graphs of square root functions and describes limitations on the domains and ranges. (B) The student relates representations of square root functions, such as algebraic, tabular, graphical, and verbal descriptions. (C) For given contexts, the student determines the reasonable domain and range values of square root functions, as well as interprets and determines the reasonableness of solutions to square root equations and inequalities. (D) The student solves square root equations and inequalities using graphs, tables, and algebraic methods. (E) The student analyzes situations modeled by square root functions, formulates equations or inequalities, selects a method, and solves problems. (F) The student expresses inverses of quadratic functions using square root functions. Rational functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on rational functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. 32 (C) Following are performance descriptions. (1) The student uses quotients to describe the graphs of rational functions, describes limitations on the domains and ranges, and examines asymptotic behavior. (2) The student analyzes various representations of rational functions with respect to problem situations. Advanced Mathematics Educational Support (f) (3) For given contexts, the student determines the reasonable domain and range values of rational functions, as well as interprets and determines the reasonableness of solutions to rational equations and inequalities. (4) The student solves rational equations and inequalities using graphs, tables, and algebraic methods. (5) The student analyzes a situation modeled by a rational function, formulates an equation or inequality composed of a linear or quadratic function, and solves the problem. (6) The student uses direct and inverse variation functions as models to make predictions in problem situations. Exponential and logarithmic functions: knowledge and skills and performance descriptions. The student formulates equations and inequalities based on exponential and logarithmic functions, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. Advanced Mathematics Educational Support Following are performance descriptions. (1) The student develops the definition of logarithms by exploring and describing the relationship between exponential functions and their inverses. (2) The student uses the parent functions to investigate, describe, and predict the effects of parameter changes on the graphs of exponential and logarithmic functions, describes limitations on the domains and ranges, and examines asymptotic behavior. (3) For given contexts, the student determines the reasonable domain and range values of exponential and logarithmic functions, as well as interprets and determines the reasonableness of solutions to exponential and logarithmic equations and inequalities. (4) The student solves exponential and logarithmic equations and inequalities using graphs, tables, and algebraic methods. (5) The student analyzes a situation modeled by an exponential function, formulates an equation or inequality, and solves the problem. 33 §111.34. Geometry (One Credit). (a) (b) Basic understandings. (1) Foundation concepts for high school mathematics. As presented in Grades K-8, the basic understandings of number, operation, and quantitative reasoning; patterns, relationships, and algebraic thinking; geometry; measurement; and probability and statistics are essential foundations for all work in high school mathematics. Students continue to build on this foundation as they expand their understanding through other mathematical experiences. (2) Geometric thinking and spatial reasoning. Spatial reasoning plays a critical role in geometry; shapes and figures provide powerful ways to represent mathematical situations and to express generalizations about space and spatial relationships. Students use geometric thinking to understand mathematical concepts and the relationships among them. (3) Geometric figures and their properties. Geometry consists of the study of geometric figures of zero, one, two, and three dimensions and the relationships among them. Students study properties and relationships having to do with size, shape, location, direction, and orientation of these figures. (4) The relationship between geometry, other mathematics, and other disciplines. Geometry can be used to model and represent many mathematical and real-world situations. Students perceive the connection between geometry and the real and mathematical worlds and use geometric ideas, relationships, and properties to solve problems. (5) Tools for geometric thinking. Techniques for working with spatial figures and their properties are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, algebraic, and coordinate), tools, and technology, including, but not limited to, powerful and accessible hand-held calculators and computers with graphing capabilities to solve meaningful problems by representing figures, transforming figures, analyzing relationships, and proving things about them. (6) Underlying mathematical processes. Many processes underlie all content areas in mathematics. As they do mathematics, students continually use problem-solving, computation in problemsolving contexts, language and communication, connections within and outside mathematics, and reasoning, as well as multiple representations, applications and modeling, and justification and proof. Geometric structure: knowledge and skills and performance descriptions. (1) 34 The student understands the structure of, and relationships within, an axiomatic system. Following are performance descriptions. (A) The student develops an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems. (B) Through the historical development of geometric systems, the student recognizes that mathematics is developed for a variety of purposes. (C) The student compares and contrasts the structures and implications of Euclidean and non-Euclidean geometries. Advanced Mathematics Educational Support (2) (3) (4) (c) The student analyzes geometric relationships in order to make and verify conjectures. The student understands the importance of logical reasoning, justification, and proof in mathematics. The student uses a variety of representations to describe geometric relationships and solve problems. Following are performance descriptions. (A) The student uses constructions to explore attributes of geometric figures and to make conjectures about geometric relationships. (B) The student makes and verifies conjectures about angles, lines, polygons, circles, and three-dimensional figures, choosing from a variety of approaches such as coordinate, transformational, or axiomatic. Following are performance descriptions. (A) The student determines if the converse of a conditional statement is true or false. (B) The student constructs and justifies statements about geometric figures and their properties. (C) The student demonstrates what it means to prove mathematically that statements are true. (D) The student uses inductive reasoning to formulate a conjecture. (E) The student uses deductive reasoning to prove a statement. Following is a performance description. The student selects an appropriate representation (concrete, pictorial, graphical, verbal, or symbolic) in order to solve problems. Geometric patterns: knowledge and skills and performance descriptions. The student identifies, analyzes, and describes patterns that emerge from twoand three-dimensional geometric figures. Advanced Mathematics Educational Support Following are performance descriptions. (1) The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles. (2) The student uses properties of transformations and their compositions to make connections between mathematics and the real world in applications such as tessellations or fractals. 35 (3) (d) Dimensionality and the geometry of location: knowledge and skills and performance descriptions. (1) (2) (e) The student analyzes the relationship between threedimensional objects and related two-dimensional representations and uses these representations to solve problems. The student understands that coordinate systems provide convenient and efficient ways of representing geometric figures and uses them accordingly. Following are performance descriptions. (A) The student describes, and draws cross sections and other slices of threedimensional objects. (B) The student uses nets to represent and construct three-dimensional objects. (C) The student uses top, front, side, and corner views of three-dimensional objects to create accurate and complete representations and solve problems. Following are performance descriptions. (A) The student uses one- and twodimensional coordinate systems to represent points, lines, line segments, and figures. (B) The student uses slopes and equations of lines to investigate geometric relationships, including parallel lines, perpendicular lines, and special segments of triangles and other polygons. (C) The student develops and uses formulas including distance and midpoint. Congruence and the geometry of size: knowledge and skills and performance descriptions. (1) 36 The student identifies and applies patterns from right triangles to solve problems, including special right triangles (45-45-90 and 30-60-90) and triangles whose sides are Pythagorean triples. The student extends measurement concepts to find area, perimeter, and volume in problem situations. Following are performance descriptions. (A) The student finds areas of regular polygons and composite figures. (B) The student finds areas of sectors and arc lengths of circles using proportional reasoning. (C) The student develops, extends, and uses the Pythagorean Theorem. (D) The student finds surface areas and volumes of prisms, pyramids, spheres, cones, and cylinders in problem situations. Advanced Mathematics Educational Support (2) (3) (f) The student analyzes properties and describes relationships in geometric figures. The student applies the concept of congruence to justify properties of figures and solve problems. Following are performance descriptions. (A) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties of parallel and perpendicular lines. (B) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of polygons and their component parts. (C) Based on explorations and using concrete models, the student formulates and tests conjectures about the properties and attributes of circles and the lines that intersect them. (D) The student analyzes the characteristics of three-dimensional figures and their component parts. Following are performance descriptions. (A) The student uses congruence transformations to make conjectures and justify properties of geometric figures. (B) The student justifies and applies triangle congruence relationships. Similarity and the geometry of shape: knowledge and skills and performance descriptions. The student applies the concepts of similarity to justify properties of figures and solve problems. Following are performance descriptions. (1) The student uses similarity properties and transformations to explore and justify conjectures about geometric figures. (2) The student uses ratios to solve problems involving similar figures. (3) In a variety of ways, the student develops, applies, and justifies triangle similarity relationships, such as right triangle ratios, trigonometric ratios, and Pythagorean triples. (4) The student describes the effect on perimeter, area, and volume when length, width, or height of a three-dimensional solid is changed and applies this idea in solving problems. Source: The provisions of this §111.34 adopted to be effective September 1, 1996, 21 TexReg 7371. Advanced Mathematics Educational Support 37 §111.35. Precalculus (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September 1, 1998, and at that time shall supersede §75.63(bb) of this title (relating to Mathematics). Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisites: Algebra II, Geometry. (b) Introduction. (c) (1) In Precalculus, students continue to build on the K-8, Algebra I, Algebra II, and Geometry foundations as they expand their understanding through other mathematical experiences. Students use symbolic reasoning and analytical methods to represent mathematical situations, to express generalizations, and to study mathematical concepts and the relationships among them. Students use functions, equations, and limits as useful tools for expressing generalizations and as means for analyzing and understanding a broad variety of mathematical relationships. Students also use functions as well as symbolic reasoning to represent and connect ideas in geometry, probability, statistics, trigonometry, and calculus and to model physical situations. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology to model functions and equations and solve real-life problems. (2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning. Students also use multiple representations, applications and modeling, justification and proof, and computation in problem-solving contexts. Knowledge and skills. (1) 38 The student defines functions, describes characteristics of functions, and translates among verbal, numerical, graphical, and symbolic representations of functions, including polynomial, rational, radical, exponential, logarithmic, trigonometric, and piecewise-defined functions. The student is expected to: (A) describe parent functions symbolically n and graphically, including y = x , x y = ln x, y = loga x, y = 1/x, y = e , x y = a , y = sin x, etc.; (B) determine the domain and range of functions using graphs, tables, and symbols; (C) describe symmetry of graphs of even and odd functions; (D) recognize and use connections among significant points of a function (roots, maximum points, and minimum points), the graph of a function, and the symbolic representation of a function; and (E) investigate continuity, end behavior, vertical and horizontal asymptotes, and limits and connect these characteristics to the graph of a function. Advanced Mathematics Educational Support (2) (3) (4) The student interprets the meaning of the symbolic representations of functions and operations on functions within a context. The student uses functions and their properties to model and solve real-life problems. The student uses sequences and series to represent, analyze, and solve real-life problems. Advanced Mathematics Educational Support The student is expected to: (A) apply basic transformations, including a • f(x), f(x) + d, f(x - c), f(b • x), |f(x)|, f(|x|), to the parent functions; B) perform operations including composition on functions, find inverses, and describe these procedures and results verbally, numerically, symbolically, and graphically; and (C) investigate identities graphically and verify them symbolically, including logarithmic properties, trigonometric identities, and exponential properties. The student is expected to: (A) use functions such as logarithmic, exponential, trigonometric, polynomial, etc. to model real-life data; (B) use regression to determine a function to model real-life data; (C) use properties of functions to analyze and solve problems and make predictions; and (D) solve problems from physical situations using trigonometry, including the use of Law of Sines, Law of Cosines, and area formulas. The student is expected to: (A) represent patterns using arithmetic and geometric sequences and series; (B) use arithmetic, geometric, and other sequences and series to solve real-life problems; (C) describe limits of sequences and apply their properties to investigate convergent and divergent series; and (D) apply sequences and series to solve problems including sums and binomial expansion. 39 (5) (6) The student uses conic sections, their properties, and parametric representations to model physical situations. The student uses vectors to model physical situations. The student is expected to: (A) use conic sections to model motion, such as the graph of velocity vs. position of a pendulum and motions of planets; (B) use properties of conic sections to describe physical phenomena such as the reflective properties of light and sound; (C) convert between parametric and rectangular forms of functions and equations to graph them; and (D) use parametric functions to simulate problems involving motion. The student is expected to: (A) use the concept of vectors to model situations defined by magnitude and direction; and (B) analyze and solve vector problems generated by real-life situations. Source: The provisions of this §111.35 adopted to be effective September 1, 1998, 22 TexReg 7623. 40 Advanced Mathematics Educational Support §111.36. Mathematical Models with Applications (One-Half to One Credit). (a) General requirements. The provisions of this section shall be implemented beginning September 1, 1998. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisite: Algebra I. (b) Introduction. (c) (1) In Mathematical Models with Applications, students continue to build on the K-8 and Algebra I foundations as they expand their understanding through other mathematical experiences. Students use algebraic, graphical, and geometric reasoning to recognize patterns and structure, to model information, and to solve problems from various disciplines. Students use mathematical methods to model and solve real-life applied problems involving money, data, chance, patterns, music, design, and science. Students use mathematical models from algebra, geometry, probability, and statistics and connections among these to solve problems from a wide variety of advanced applications in both mathematical and nonmathematical situations. Students use a variety of representations (concrete, numerical, algorithmic, graphical), tools, and technology to link modeling techniques and purely mathematical concepts and to solve applied problems. (2) As students do mathematics, they continually use problem-solving, language and communication, connections within and outside mathematics, and reasoning. Students also use multiple representations, applications and modeling, justification and proof, and computation in problem-solving contexts. Knowledge and skills. (1) (2) The student uses a variety of strategies and approaches to solve both routine and non-routine problems. The student uses graphical and numerical techniques to study patterns and analyze data. Advanced Mathematics Educational Support The student is expected to: (A) compare and analyze various methods for solving a real-life problem; (B) use multiple approaches (algebraic, graphical, and geometric methods) to solve problems from a variety of disciplines; and (C) select a method to solve a problem, defend the method, and justify the reasonableness of the results. The student is expected to: (A) interpret information from various graphs, including line graphs, bar graphs, circle graphs, histograms, and scatterplots to draw conclusions from the data; (B) analyze numerical data using measures of central tendency, variability, and correlation in order to make inferences; (C) analyze graphs from journals, newspapers, and other sources to determine the validity of stated arguments; and 41 (D) (3) (4) (5) (6) 42 The student develops and implements a plan for collecting and analyzing data in order to make decisions. The student uses probability models to describe everyday situations involving chance. The student uses functional relationships to solve problems related to personal income. The student uses algebraic formulas, graphs, and amortization models to solve problems involving credit. use regression methods available through technology to describe various models for data such as linear, quadratic, exponential, etc., select the most appropriate model, and use the model to interpret information. The student is expected to: (A) formulate a meaningful question, determine the data needed to answer the question, gather the appropriate data, analyze the data, and draw reasonable conclusions; (B) communicate methods used, analysis conducted, and conclusions drawn for a data-analysis project by written report, visual display, oral report, or multimedia presentation; and (C) determine the appropriateness of a model for making predictions from a given set of data. The student is expected to: (A) compare theoretical and empirical probability; and (B) use experiments to determine the reasonableness of a theoretical model such as binomial, geometric, etc. The student is expected to: (A) use rates, linear functions, and direct variation to solve problems involving personal finance and budgeting, including compensations and deductions; (B) solve problems involving personal taxes; and (C) analyze data to make decisions about banking. The student is expected to: (A) analyze methods of payment available in retail purchasing and compare relative advantages and disadvantages of each option; (B) use amortization models to investigate home financing and compare buying and renting a home; and Advanced Mathematics Educational Support (C) (7) (8) (9) The student uses algebraic formulas, numerical techniques, and graphs to solve problems related to financial planning. The student uses algebraic and geometric models to describe situations and solve problems. The student uses algebraic and geometric models to represent patterns and structures. use amortization models to investigate automobile financing and compare buying and leasing a vehicle. The student is expected to: (A) analyze types of savings options involving simple and compound interest and compare relative advantages of these options; (B) analyze and compare coverage options and rates in insurance; and (C) investigate and compare investment options including stocks, bonds, annuities, and retirement plans. The student is expected to: (A) use geometric models available through technology to model growth and decay in areas such as population, biology, and ecology; (B) use trigonometric ratios and functions available through technology to calculate distances and model periodic motion; and (C) use direct and inverse variation to describe physical laws such as Hook's, Newton's, and Boyle's laws. The student is expected to: (A) use geometric transformations, symmetry, and perspective drawings to describe mathematical patterns and structure in art and architecture; and (B) use geometric transformations, proportions, and periodic motion to describe mathematical patterns and structure in music. Source: The provisions of this §111.36 adopted to be effective September 1, 1998, 22 TexReg 7623. Advanced Mathematics Educational Support 43 Subchapter D. Other High School Mathematics Courses Statutory Authority: The provisions of this Subchapter D issued under the Texas Education Code, §28.002, unless otherwise noted. §111.51. Implementation of Texas Essential Knowledge and Skills for Mathematics, Other High School Mathematics Courses. The provisions of this subchapter shall be implemented by school districts beginning September 1, 1998, and at that time shall supersede §75.63(o), (q)-(u), and (cc) of this title (relating to Mathematics). Source: The provisions of this §111.51 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.52. Independent Study in Mathematics (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of Independent Study in Mathematics. Required prerequisites: Algebra II, Geometry. Students may repeat this course with different course content for a second credit. (b) Content requirements. Students will extend their mathematical understanding beyond the Algebra II level in a specific area or areas of mathematics, such as theory of equations, number theory, nonEuclidean geometry, advanced survey of mathematics, or history of mathematics. The requirements for each course must be approved by the local district before the course begins. (c) If this course is being used to satisfy requirements for the Distinguished Achievement Program, student research/products must be presented before a panel of professionals or approved by the student's mentor. Source: The provisions of this §111.52 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.53. Advanced Placement (AP) Statistics (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisites: Algebra II, Geometry. (b) Content requirements. Content requirements for Advanced Placement (AP) Statistics are prescribed in the College Board Publication Advanced Placement Course Description: Statistics, published by The College Board. This publication may be obtained from the College Board Advanced Placement Program. Source: The provisions of this §111.53 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.54. Advanced Placement (AP) Calculus AB (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisite: Precalculus. (b) Content requirements. Content requirements for Advanced Placement (AP) Calculus AB are prescribed in the College Board Publication Advanced Placement Course Description Mathematics: Calculus AB, Calculus BC, published by The College Board. This publication may be obtained from the College Board Advanced Placement Program. Source: The provisions of this §111.54 adopted to be effective September 1, 1998, 22 TexReg 7623. Advanced Mathematics Educational Support 45 §111.55. Advanced Placement (AP) Calculus BC (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of this course. Recommended prerequisite: Precalculus. (b) Content requirements. Content requirements for Advanced Placement (AP) Calculus BC are prescribed in the College Board Publication Advanced Placement Course Description: Calculus AB, Calculus BC, published by The College Board. This publication may be obtained from the College Board Advanced Placement Program. Source: The provisions of this §111.55 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.56. IB Mathematical Studies Subsidiary Level (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of IB Mathematical Studies Subsidiary Level. To offer this course, the district must meet all requirements of the International Baccalaureate Organization, including teacher training/certification and IB assessment. Recommended prerequisites: Algebra II, Geometry. (b) Content requirements. Content requirements for IB Mathematical Studies Subsidiary Level are prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from International Baccalaureate of North America. Source: The provisions of this §111.56 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.57. IB Mathematical Methods Subsidiary Level (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of IB Mathematical Methods Subsidiary Level. To offer this course, the district must meet all requirements of the International Baccalaureate Organization, including teacher training/certification and IB assessment. Recommended prerequisites: Algebra II, Geometry. (b) Content requirements. Content requirements for IB Mathematical Methods Subsidiary Level are prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from International Baccalaureate of North America. Source: The provisions of this §111.57 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.58. IB Mathematics Higher Level (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of IB Mathematics Higher Level. To offer this course, the district must meet all requirements of the International Baccalaureate Organization, including teacher training/certification and IB assessment. Recommended prerequisite: IB Mathematical Studies Subsidiary Level or IB Mathematical Methods Subsidiary Level. (b) Content requirements. Content requirements for IB Mathematics Higher Level are prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from International Baccalaureate of North America. Source: The provisions of this §111.58 adopted to be effective September 1, 1998, 22 TexReg 7623. 46 Advanced Mathematics Educational Support §111.59. IB Advanced Mathematics Subsidiary Level (One-Half to One Credit). (a) General requirements. Students can be awarded one-half to one credit for successful completion of IB Advanced Mathematics Subsidiary Level. To offer this course, the district must meet all requirements of the International Baccalaureate Organization, including teacher training/certification and IB assessment. Recommended prerequisite: IB Mathematics Higher Level. (b) Content requirements. Content requirements for IB Advanced Mathematics Subsidiary Level are prescribed by the International Baccalaureate Organization. Curriculum guides may be obtained from International Baccalaureate of North America. Source: The provisions of this §111.59 adopted to be effective September 1, 1998, 22 TexReg 7623. §111.60. Concurrent Enrollment in College Courses. (a) General requirements. Students shall be awarded one-half credit for each semester of successful completion of a college course in which the student is concurrently enrolled while in high school. (b) Content requirements. In order for students to receive state graduation credit for concurrent enrollment courses, content requirements must meet or exceed the essential knowledge and skills in a given course. Source: The provisions of this §111.60 adopted to be effective September 1, 1998, 22 TexReg 7623. Advanced Mathematics Educational Support 47 APPENDIX B TOPIC OUTLINES FOR ADVANCED PLACEMENT MATHEMATICS The following general topic outlines for AP Calculus AB, AP Calculus BC, and AP Statistics are adapted from the College Board course outlines for each of the courses. More detailed course descriptions are available from the College Board.34 AP Calculus AB Functions, Graphs and Limits • • • • • • • Analysis of graphs Understanding the concept of a limit Calculating limits Limits involving infinity Asymptotic and unbounded behavior Continuity Implications of continuity Differentiation • • • • • • • • • • • • • • Concept of the derivative Derivatives of basic functions Product and quotient rules Chain rule Implicit differentiation Mean value theorem Corresponding characteristics of f and f' Geometric interpretation of the second derivative Corresponding characteristics among f, f', and f" Tangent lines as approximations Optimization Analysis of curves given in Cartesian form Modeling rates of change Related rates Integration • • • • • • • The Riemann Sum The definite integral Fundamental Theorem of Calculus Antidifferentiation techniques Properties of integrals Accumulation of rate of change Numerical approximations of definite integrals 34 To download the Advanced Placement course descriptions, go to apcentral.collegeboard.com/courses/descriptions and choose from the table of AP courses at the bottom of the page. 48 Advanced Mathematics Educational Support • • • • Applications of the integral Separable differential equations Initial value problems Exponential growth and decay AP Calculus BC Functions, Graphs and Limits • • • • • • • Analysis of graphs Understanding the concept of a limit Calculating limits Limits involving infinity Asymptotic and unbounded behavior Continuity Implications of continuity Differentiation • • • • • • • • • • • • • • • • • Concept of the derivative Derivatives of basic functions Derivatives of parametric, polar, and vector functions Product and quotient rules Chain rule Implicit differentiation Mean value theorem Corresponding characteristics of f and f' Geometric interpretation of the second derivative Corresponding characteristics among f, f', and f" Tangent lines as approximations Optimization Analysis of curves given in Cartesian form Analysis of curves given in parametric, polar, or vector forms Modeling rates of change Related rates L’Hopital’s Rule Integration • • • • • • • • • • • The Riemann Sum The definite integral Fundamental Theorem of Calculus Antidifferentiation techniques Properties of integrals Accumulation of rate of change Numerical approximations of definite integrals Applications of the integral Separable differential equations Logistic differential equations Initial value problems Advanced Mathematics Educational Support 49 • • • • Slope fields Euler’s method Exponential growth and decay Improper integrals Sequences and Series • • • • Concept of series Special series Tests for convergence Taylor series AP Statistics Exploring Data • • • • • • • • • • Describing distributions with graphs Numerical summaries of distributions Boxplots Comparing distributions Measures of relative position Correlation Linear regression Model quality Transformations to achieve linearity Analyzing two-way tables Planning a Study • • • Simple random sampling Stratified, cluster and systematic sampling Experimental design Anticipating Patterns • • • Random variables Normal distribution Sampling distributions Statistical Inference: Confirming Models • • • • • • • • • 50 Confidence intervals Estimating population parameters Tests of significance Type I/II errors and power Difference in means Difference in proportions Chi-Square test for goodness of fit Inference for two-way tables Inference for slope of the least squares line Advanced Mathematics Educational Support APPENDIX C VERTICAL TEAMS: A STRATEGY FOR BUILDING SCHOOL CAPACITY Higher Education Involvement in Vertical Teaming Vertical teaming is a vehicle through which higher education mathematics faculty can understand and influence the secondary school mathematics experience of students. Ideally, the gateway to the full range of mathematics courses should remain open for every student as long as possible. The courses students take as early as the sixth grade often determine the sequence of mathematics courses available to them throughout high school. In fact, the mathematics curriculum that students take beginning in kindergarten determines the mathematical content they can pursue in secondary school as well as at the collegiate level. Although not every student will go on to pursue a mathematics-based field, all students will be stronger for the experiences of a challenging and rigorous four-year secondary mathematics program. The mathematics and analytical skills they learn will be useful to them as citizens and in whatever fields they choose to pursue. Moreover, higher expectations for themselves in mathematics can enhance student self-esteem and may translate to higher expectations in other aspects of their lives as well. To succeed in giving all students access to a challenging and rigorous secondary mathematics program, all teachers K–16 must communicate and work with one another. A key strategy to foster such collaboration is the formation of K–16 vertical teams. The primary goal of the vertical team strategy is to enhance all students’ achievement by increasing communication and cooperation among the members of the teaching team about the mathematics program at their schools. Through such communication and cooperation, vertical teams can facilitate the implementation of academic changes and support structures necessary to make high achievement in mathematics by all students a reality. Higher Education Leadership in Vertical Teaming Since vertical teams are already working in some Texas districts, the first item of business is for higher education mathematics faculty to determine whether a vertical team is already in place in the district with which they wish to work. Vertical Teaming Not in Place If vertical teaming is not in place, then the higher education faculty member can work to initiate interest. In other words, the higher education faculty member must sell the local decision-makers on the need and benefits of vertical teaming. A suggested sequence of events follows. Step 1: Contact the local district superintendent. The superintendent should understand the benefits of vertical teaming for the faculty and students in the district, including the impact on students, teachers, and the districts’ accountability rating. If the district has not been involved in vertical teaming, then initial involvement in one content area is strongly suggested, with mathematics as this area. Step 2: Ask the superintendent for permission to visit with the high school administration about the concept of vertical teaming. Again, the campus administrators will need to see Advanced Mathematics Educational Support 51 the benefits for implementing vertical teaming on their campus. The benefits for the campus will include the same benefits as for the district. Step 3: Ask the campus administrators for permission to visit with the mathematics faculty about implementing vertical teams on the campus. Ideally, the campus administrators should attend this meeting. The benefits for the teachers and their students are discussed at this meeting. One strong benefit for the teachers is that they will receive students who are better prepared to pursue challenging and rigorous mathematics. Step 4: Visit feeder schools35 to discuss the vertical team concept with the feeder schools’ campus administrators and mathematics faculty. Hopefully, the vertical team concept can be backed down to the kindergarten level; thus, the team initially may include teachers from grades K–12 and the higher education mathematics faculty members. Step 5: Work with district leadership to select members of the vertical team. The vertical team should involve interested teachers from each grade level and/or mathematics course on each campus. Past experience has indicated that the number of members on a successful vertical team should be between 10 and 20. Each individual affected by the vertical team can provide input to the team’s work, but a vehicle should be established for such input, rather than have an unwieldy number of members on the vertical team. In addition to mathematics teachers, members of the vertical team should include the following: • At least one member who has some degree of expertise related to the use of technology in mathematics education. • At least one member who has a deep, extensive knowledge of college-level mathematics. Efforts to create a strong mathematics program must include serious attention to the mathematics itself, and if students are to be adequately prepared to succeed in postsecondary mathematics, the team will need a picture of what postsecondary mathematics looks like. • Administrators from the schools represented on the vertical team, particularly in discussions of inclusiveness and student support structures, access to advanced mathematics courses, professional development, and local action planning. • Counselors, especially in discussions of inclusiveness and student support structures, access to advanced mathematics courses, and local action planning. To be maximally effective, the mathematics vertical team will need support from campus administrators at each of the schools represented by the team. Critical roles played by administrators include the following: • Making the establishment of a strong, inclusive mathematics program a top priority for the school; • Assisting with shaping the team’s goals and vision of where their efforts are headed; 35 A feeder school is a school whose students attend the next grade level at another campus. For example, a certain middle school may “feed into” one designated high school. 52 Advanced Mathematics Educational Support • Communicating with district administration, teachers in other departments, the school board, parents, and community members about the goals of the mathematics vertical team; explaining why the team’s efforts are important; helping others to value the team’s efforts; and defusing any concerns about the team and their efforts; • Examining how the team’s work will benefit the entire school and building the team’s work into a schoolwide improvement plan or strategic plan; • Providing strong visible support for the team’s efforts. Administrators need to help with a number of logistical issues. Administrators should not expect vertical teams to meet on the teachers’ own time without compensation; not only is such an expectation unfair, it also defeats the purpose of the formalization of communication structures enabled by vertical teaming. Many teachers do not wish to be away from their classes during the school day, even if the team meets only 4 to 5 times a year. Additionally, since the vertical team will typically involve several campuses, travel time must be considered in scheduling time for team meetings. If a viable time to meet cannot be identified during the school day, then the higher education faculty can help in identifying grants to fund teachers’ extra hours. Higher education faculty can also be a valuable resource in writing proposals for funding projects. If school district or grant funds are not available, then a local sponsor might be identified for the vertical team efforts. The local sponsor may be an individual or business partnership with an interest in mathematics preparation of the local youth. After members of the vertical team are identified and time to meet is identified, the team must develop into a cohesive unit rather than a group of individuals. This development of a cohesive unit is much the same as that done by a coach as he or she takes a group of individual athletes and builds them into a team. Each member must develop the attitude that the team’s interest is of greater value than any one individual’s interest. Vertical Teaming Already in Place If a functioning vertical team is already in place in a district or campus, higher education mathematics faculty face different challenges. One major issue can be that the higher education mathematics faculty member has to integrate him- or herself into an already established team. However, this is an issue not only with the addition of a higher education faculty member to the team, but also as any new members are added to the vertical team as a result of retirement, resignation, or personal reasons. Periodically, the vertical team will need to experience additional team-building activities to orient new members. If a team is already in place and the higher education faculty member desires to become involved, then the faculty member should go through a process similar to that described above. However, if the team is already functioning, then the faculty member will not need to “sell” the district on the concept but instead will need to share with the various administrators and teachers the benefits that can occur from the involvement of higher education faculty, such as improved communication between higher education and the local school system and better preparation for students who leave the school system to attend a college or university. Advanced Mathematics Educational Support 53 Vertical Alignment of the K–16 Curriculum The vertical alignment of the curriculum process K–16 operates similarly to the process in K–12. To include higher education, one of the first steps for vertical team is to decide what higher education courses the team should focus on. That is, what higher education courses should the team connect vertically with high school mathematics courses, by identifying the mathematical content needed for success in those higher education courses? The higher education faculty can identify the content necessary for success in the higher education courses. They can also guide the secondary mathematics teachers regarding what content needs to be covered as a prerequisite for success in those higher education courses. Some vertical teams may wish to consider a combination of College Algebra and Trigonometry, while others may consider the calculus sequence. After selecting the courses, the higher education mathematics faculty identifies critical skills and knowledge that all students need to have to succeed in that higher education course. Next, the team identifies where that content is taught and learned in the secondary curriculum. Then, the highest-level relevant mathematics course in the high school curriculum is investigated. The high school teachers should identify the skills and knowledge that the students need to be successful in those high school courses. After this content is identified, then the team identifies where that content is taught and learned in the secondary curriculum as described in the TEKS. The high school faculty can provide information on why some of the content may or may not be appropriate for the high school course. Working together as a team, the high school and higher education faculty can reach consensus on the appropriate content at each level and the best tools for teaching that content for the success of all students. As collaborations develop that promote coordination of precollege and postsecondary mathematics teaching, one outcome will be enhanced capacity at the high school level to prepare more students to immediately enroll in the more challenging mathematics courses that they need for success. 54 Advanced Mathematics Educational Support REFERENCES Adelman, C. 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