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CERT Mathematics Standards for California High School Graduates

Over the last decade, each segment of California education has faced profound policy questions that have at their root the compelling need to improve the quality of education for all students at all levels. Confidence in the public school system is being severely tested by the low performance of California students on standardized examinations, particularly in reading, writing, and mathematics. In response to this concern, Californians overwhelmingly support setting higher goals for students, and are convinced that higher standards will lead to increased student learning. Further, they believe there should be specific guidelines for what students should know and be able to do.¹

The K–12 segment is responding by focusing greater attention on increasing student learning so that all students have increased opportunities to pursue postsecondary education or a career upon graduating from high school. This improvement of academic preparation of high school graduates is central to the California State University’s efforts to reduce the need for remedial education in the CSU system; the University of California’s need for an expanded, diverse pool of fully prepared high school graduates; and the California community college system’s goal of preparing more students for transfer to four year institutions and for employment in technical fields. It is clear that each education segment’s ability to fulfill its mission depends greatly on the success of the other segments. There is an unequivocal understanding that all levels of education, kindergarten through college, must work together. As the Education Round Table recently stated:

“Not only are resources and capacities stretched by competing interests and priorities, but also the problems themselves are inextricably tied to common interests and responsibilities. Now more than ever, we must plan and work together in integrated, focused ways to ensure an acceptable level of academic success for all students, thereby providing equal access to opportunities for higher education, meaningful employment, and full participation in our economy and democratic society.”²

It is within that context and with that spirit of cooperation that the Mathematics Graduation Standards Task Force began its work. In this section, we outline the recent history and charge to the task force, discuss standards generally, and deal with the important differences between these standards and what students are expected to know and be able to do to succeed in college. We close this section by describing the characteristics of high school students who have mastered the content standards that are set forth in section two of the report. Section three focuses on important implementation issues.

A Call to Action
In 1995, California State Superintendent of Public Instruction Delaine Eastin appointed a broad-based task force to make recommendations to improve the quality of mathematics performance in California’s schools. Its recent report, “A Call to Action: Improving Mathematics Achievement for All California Students,” recommended the immediate adoption of clear and specific content and performance standards for mathematics. Specifically, the panel recommended that graduation standards be adopted “to ensure that standards for high schools reflect what students are expected to know and be able to do as a condition of receiving a high school diploma.”³ The report further called for “a balanced program in content, which includes basic skills, conceptual understanding, and problem solving involving a variety of strategies learned from direct instruction and exploration.”⁴

Mathematics Graduation Standards Task Force
In 1996, the California Education Round Table⁵ appointed the Mathematics Graduation Standards Task Force, made up of a broad cross-section of Californians, including classroom teachers, school and district administrators, university and college faculty, and community and business leaders concerned with mathematics learning achievement in California schools. Our charge was to agree upon clear and specific mathematics content standards for high school graduation.

The focus of the Mathematics Graduation Standards Task Force is on the knowledge and skills students are able to display upon graduation from high school. However, the task force realizes that sound preparation from elementary through middle school is essential if students are to take the high school mathematics courses necessary to meet the rigorous graduation standards set forth in this document. Standards agreed upon by the task force are important in two contexts. First, they will contribute to the standards-setting process described in AB 265, which calls for the establishment of a Commission on Academic Content and Performance Standards. The Commission is required to first address reading, writing and mathematics at all grade levels 1–12, to be followed by other subject matter areas. The results of the work of the Mathematics Graduation Standards Task Force will be submitted to the Commission and are expected to have significant impact on the Commission’s recommendations concerning grade 9–12 standards. The Commission is then to provide recommendations to the State Board of Education for adoption.

Secondly, the task force will contribute to building a common understanding among school teachers, college and university professors, school board members, administrators, employers, and parents about the mathematics skills, competencies, and knowledge that students need to acquire in high school in order to graduate and successfully progress to the next level of their lives, whether it be directly into a career, further career preparation, or postsecondary education.

At the outset, it is important to note that the goal of the standard setting process is to encourage higher achievement—not just higher standards. Unless substantial numbers of students meet the standards, the overall objective, which is to broaden the opportunities available to them, will not be reached. Achieving a high level of mathematics proficiency will open opportunities to students who wish to pursue jobs in highly technical fields, and to those who wish to pursue a wider array of postsecondary opportunities.

Content Standards and Performance Standards
In addressing the charge from the Round Table to agree upon standards for high school–level mathematics, the task force has been guided by the proposition that content standards must be connected to performance standards. One is meaningless without the other. Content standards define “what” teachers are expected to teach, and what students are expected to learn. Performance standards define the degrees to which students do learn, with higher degrees of mastery expected, for example, of students planning to be majors in science and mathematics in universities than those for high school graduates generally.⁶ In addition, an increasing number of Californians will opt for vocational-technical training. The curricula of these programs are rapidly increasing in complexity and difficulty as careers become more technical. Therefore, candidates for entry and ultimate success in these programs must complete rigorous mathematical coursework at or above baseline performance standards. The content standards contained in this document are intended to be a part of a strong foundation for all students, irrespective of their ultimate career goals or aspirations.

Balanced Standards
Throughout our deliberations, the task force has grappled with the appropriateness of each standard. If we raise the standard unrealistically high, the end result could be more student failure, and teacher and parent discouragement. If we don’t raise the standard high enough, the end result will be the continued failure to prepare our students adequately for their futures. We have sought to achieve an appropriate balance, arriving at high, but attainable standards. Our guiding question has been: What is absolutely essential for a mathematically literate high school graduate to know and be able to do?

On each of the standards we have sought consensus across the broad cross-section of views entertained by task force members. We have concluded that the standards suggested here represent a marked improvement over existing standards. Currently, standards are adopted on a district-by-district level and represent a much lower degree of achievement—much closer, in most districts, to an eighth grade (or even lower) level of proficiency.

Curriculum Options
These standards define what students should learn in order to graduate from high school; the standards do not define a particular sequence of courses. Schools should use these standards to reexamine course content and sequences. Some schools may decide to enroll most students in a common two or three year sequence to complete achievement of these standards. Many students would go on in subsequent courses to complete study that meets expectations for entering college freshmen. Other schools will offer options to students for courses that differ in rigor and pace, but have the achievement of these standards in common. In any case, all students will have open longer the option of preparing for college; and all students will better be prepared for a future increasingly impacted by technology.

Graduation Standards and College-Bound Students
Another important charge to the task force from the Education Round Table, was to “clarify the relationship between these graduation standards and expected competencies for entering freshmen.” The Intersegmental Council of Academic Senates (ICAS) is in the process of adopting a revised “Statement of Competencies in Mathematics Expected of Entering College Students,” which spells out what students need to know and be able to do to be successful in college. That document appropriately sets forth expectations for college–bound students. This document is intended to convey mathematics expectations for all students.

The difference between what all high school graduates are currently expected to know and be able to do in mathematics, and what is expected for postsecondary-bound students is enormous. The standards presented in this report move the expectations for all high school graduates substantially closer to the expectations for college-bound students. There remains, however, a gap between the two, and it is important to articulate what the differences are. First, students who expect to enter a four year higher education institution are expected to have completed at least three years of rigorous college preparatory mathematics instruction. Indeed, most students who are admitted to highly selective institutions have completed four or more years of college preparatory mathematics. These standards require a rigorous course of study incorporating core concepts and skills from algebra I and geometry, and include data analysis. Courses designed to prepare students to meet these standards are appropriate for students who intend to continue to study mathematics in preparation for college, as well as students who choose not to study mathematics beyond the content specified in the standards.

Secondly, a higher level of performance is expected of college-bound students. All college-bound students are expected to have greater facility and fluency with the basic techniques of mathematics, a deeper understanding of the underlying concepts and the logical reasoning that is central to mathematics. College-bound students should have the ability to solve more sophisticated problems and to use mathematics in a greater variety of applications.

Student Characteristics
Standards are proposed which the task force believes will help develop the following characteristics: High school graduates should have acquired a fluency in fundamental mathematical skills and their use, and realize that they can continue to develop and perfect new skills throughout their study of mathematics at all grade levels. Graduates should also attain the perspective that mathematics provides a way of understanding, with its own structure, and is not just a sequence of algorithms or manipulations of symbols to be used without reflection. Graduates should see mathematics as a way of helping them to understand the world in which they live. Graduates should be able to solve problems that arise both in concrete and abstract situations, including problems that require time and thought, and that go beyond mimicking familiar examples. Graduates should be able to recognize patterns, make conjectures, test these conjectures, and should understand that mathematical assertions require justification based on persuasive arguments.

Adoption of these standards will have enormous implications for all of education, but especially for high schools. In the next section we examine key issues related to implementation.

IMPLEMENTATION

In this section we explore the impact that adoption of these standards will have on high school mathematics instruction, and on the use of technology in schools—both important implementation issues. In addition, we discuss the kinds of systemic commitment required to make these new standards effective.

Mathematics Instruction
As the Intersegmental Council of the Academic Senates concluded in their statement of Competencies:

“There is no best approach to teaching, not even an approach that is effective for all students, or for all instructors. One criterion that should be used in evaluating approaches to teaching mathematics is the extent to which they lead to the development in the student of the dispositions, concepts, and skills that are crucial to success. It should be remembered that in the mathematics classroom, time spent focused on mathematics is crucial. The activities and behaviors that can accompany the learning of mathematics must not become goals in themselves—understanding of mathematics is always the goal.”⁷

The report identifies the kinds of activities that should be promoted in the classroom. We include them here in summary form:

• Becoming Fluent in Mathematics - Effective classes will provide experiences that lead to students’ acquisition of facility with the basic techniques of mathematics. There are certain necessary skills that students will need to be able to call upon without hesitation.

• Modeling Mathematical Thinking - Effective classes will reflect the enthusiasm that comes in learning with a teacher and others who get excited about mathematics, who work as a team, who experiment and form conjectures.

• Solving Problems - Effective classes will reflect the notion that problem solving is best conveyed by giving students appropriate experience in solving unfamiliar problems, by then engaging them in discussion of their various attempts at solutions, and by reflecting on these processes.

• Developing Analytic Ability and Logic - Effective teachers will emphasize a thorough understanding of the subject matter and the development of logical reasoning. A classroom full of discourse and interaction that focuses on reasoning is a classroom in which analytic ability and logic are being developed.

• Experiencing Mathematics in Depth - Students must delve deeply into well-chosen areas of mathematics. A shallow exploration of a broad range of topics will not contribute to the ability to understand and independently use mathematics.

• Appreciating the Beauty and Fascination of Mathematics - Effective teachers must nurture the appreciation for the inherent beauty of mathematics.

• Building Confidence - Effective teachers must create situations in which students are rewarded for being inquisitive, for experimenting, for taking risks, and for persisting in finding solutions they fully understand. This will help students generate confidence in their mathematical ability.

• Communicating - Effective classes will reflect a student’s ability to explain with confidence not only the answer to the problem but the process by which the problem was solved. Students need extensive experience in oral and written communication with extensive feedback in order to develop these skills.

While the development of these dispositions in the student is important, the larger goal is always student understanding of mathematics. As a student develops a mathematical skill, be it a basic skill or a more advanced skill, the development of an understanding is crucial in order that the skill be lasting and one that the student can apply in different contexts. As the ICAS document states:

“Then, as they use the skill in different contexts, they gradually wean themselves from thinking about it deeply each time, until its application becomes routine. But their understanding of the mathematics is the map they use whenever they become disoriented in this process. The process of applying skills in varying and increasingly complex applications is one of the ways that students not only sharpen their skills, but also reinforce and strengthen their understanding. Thus, in the best of mathematical environments, there is no dichotomy between gaining skills and gaining understanding. A curriculum that is based on depth and problem solving can be quite effective in this regard provided that it focuses on appropriate areas of mathematics.”⁸

Use of Technology
Just as there is no “best” approach to teaching mathematics, there is no fail-safe rule which tells us whether or when it is appropriate to use technology to convey the important concepts contained in the standards. Technology is not a substitute for mastery. Neither is it a panacea for the shortcomings of mathematics instruction. It is a tool which, when used appropriately, can enhance mathematics instruction. In making judgments about its appropriateness, teachers and learners must always keep sight of the goal—understanding mathematics.

Educators should actively explore ways that technology can support students in their progress toward achieving understanding of the basic ideas, skills, and techniques represented by these mathematics standards. Many jobs of the future will require the creative use of technology to explore data, analyze information, and solve problems. Opportunities to use technology can assist students in learning basic skills, modeling mathematical thinking, developing analytical ability and logic, and in communicating their understanding of important mathematics concepts. “Technology should not be used just because it is appealing. But it must be used when it can enhance the teaching and learning of mathematics.”⁹

The roles of calculators and computers in learning, creating, and applying mathematics are changing almost as quickly as computational technology is advancing. In helping students to learn mathematics, care must be taken in using emerging technologies. Appropriate use might provide students with more access to deeper understanding but caution is needed to prevent rushing to technology in ways that deny students the opportunity to acquire certain skills that can assist their understanding of mathematical concepts and structures. For example, students must be given ample opportunities to master arithmetic computation with whole numbers and fractions without calculators, both in order to be able to perform quick computations with a minimum of effort and in order to develop an internal sense of the meaning and results of such computations through their own experience. On the other hand, technology allows us to perform many more computations with great accuracy and so should be available to students to allow study of realistic data without the burden of performing many time-consuming calculations.

Opportunity to Succeed—a Shared Responsibility
Increasingly, reformers have come to recognize that merely raising standards is not enough to raise student achievement. Improved curriculum, better prepared teachers, and changes in the organization and management of schools are also necessary to bring about improvement in student performance. The teaching and learning of mathematics should not be limited to formal classes in mathematics. Students will be motivated to learn and mathematics will come alive if examples and applications are available throughout the curriculum. The physical sciences have been the traditional source for applications in mathematics. But virtually every course in the high school curriculum provides opportunities to use mathematics. The social sciences are especially rich in examples in the areas of data analysis and mathematical reasoning. Newspapers and news magazines are readily accessible sources for applications.

Ensuring that students are able to meet these higher standards will require a substantial commitment by all those associated with education. State leadership must provide clear, unequivocal expectations of what students are expected to know and be able to do. Schools must engage parents and the community to help students reach the new standards. Parents need to know and understand early on, even as early as the primary grades, the increased expectations required to attain a high school diploma. Parents should become familiar with the school’s mathematics curriculum, engage in a continual dialogue with school personnel regarding their child’s achievement levels in mathematics, and provide continuous support and encouragement in the area of mathematics. Students must be prepared to work harder—they must accept the responsibility for their own learning. “Students should realize that their minds are their most important mathematical resource, and that teachers and other students can help them to learn but can’t learn for them.”¹⁰ Prospective employers and postsecondary institutions must take into account more than grades or receipt of a diploma in assessing readiness for employment or collegiate studies. Mathematics departments and schools of education must collaborate in order to increase their emphasis on the strong mathematical preparation of prospective teachers. Furthermore, teacher preparation programs must expect new teachers to understand the importance of these mathematics content standards, to rethink their role as effective teachers, and to change their teaching practices so that all their students meet these standards. School districts must demand that prospective new teachers have the necessary mathematics background to teach adequately to the new standards. In turn, communities need to hold schools accountable for providing the educational environment that will foster student success.

Conclusion
These mathematics standards for high school graduates have been prepared by task force members, motivated by a spirit of cooperation and consensus rarely experienced among representatives of the different segments of education. Our work has been enriched by thoughtful and sound contributions from parents, employers, and community leaders. We have taken our charge from the California Education Round Table most seriously because we believe that changes must occur in the teaching and learning process if the young people of today and tomorrow are to have the skills necessary to compete successfully and prosper in a world and in an economy that is directed by the power of information rather than simply industrial might.

We urge school boards, state and local policy-makers, employers, parents, and the general public to consider these standards closely and to see that they are reflected in the curricula of their local schools. Further, we strongly support university and college efforts to work more closely with the schools in accomplishing the changes necessary to implement these content standards rapidly and pervasively. We must move forward together with purpose and dedication.

REFERENCES

1 “Californians’ Views on Education: Results of the 1996 PACE Poll,” Policy Analysis for California Education, Berkeley, CA, April 1996.

2 “Joint Statement on Collaborative Initiatives to Improve Student Learning and Academic Performance, Kindergarten through College,” Education Round Table

3 “A Call to Action: Improving Mathematics for All California Students,” California Department of Education, Sacramento, CA, 1995, page 5.

4 Ibid., page 6

5 The California Education Round Table is a voluntary organization consisting of the Superintendent of Public Instruction, the President of the University of California, the Chancellor of the California State University, the Chancellor of the California Community Colleges, a representative of California Independent Colleges and Universities and the Executive Director of the California Postsecondary Education Commission.

6 In the section on standards, we take a first step in setting those performance standards by citing exemplars which suggest approaches for developing performance standards based upon these content standards.

7 “Statement on Competencies in Mathematics Expected of Entering College Students,” Intersegmental Council of Academic Senates, Final Draft, 1996, page 3.

8 Ibid., page 5.

9 Mathematics Sciences Education Board of the National Resource Council, Reshaping School Mathematics: A Philosophy and Framework of Curriculum, 1990, page 37.

10 “Statement on Competencies in Mathematics Expected of Entering College Students,” page 2.

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