The State Board of Education adopted the Ohio Academic Content Standards for K-12 Mathematics in December 2001. The Standards set high learning expectations for every student, recognizing that in the 21st century, every student will need a strong preparation in mathematics. In Ohio, the assumption is that all students can learn significant mathematics, and the commitment is that all students will be successful in learning mathematics and will graduate from high school fully prepared for the demands of the workplace and further study.

Many factors influence how secondary mathematics programs can best be designed and delivered at this time. Day-to-day decision making, as well as the expectations for today’s workforce, require a greater emphasis on data analysis, probability, and statistics in the secondary curriculum. The tools of technology make some mathematical concepts accessible to students at an earlier stage. The curriculum of the middle grades now includes many of the basic concepts of algebra, geometry, measurement, and data analysis. Consequently, what is needed in many Ohio districts is not a simple adjustment on the margin of an old curriculum, but rather a full rethinking of the secondary school mathematics program.

There are many ways a curriculum can be configured to respond to the requirements of the Content Standards. In the area of secondary mathematics, the Department of Education is providing districts with three different models for mathematics programs in grades 9-12.

The three models were drafted by a panel of Ohio teachers, mathematicians, and mathematics educators in the summer of 2005. They were reviewed and discussed by professional groups, practitioners, and others during the school year 2005-2006, and after revision, are now available to schools. The models are presented in terms of years of study (Year 1 through Year 5) rather than in terms of grade levels (grade 9 through grade 12), recognizing that some students will start the secondary mathematics curriculum in grade 8, others in grade 9, and that there can be years when some students take more than one mathematics course. The models emphasize the importance of every student taking mathematics in each of the four years of high school, and they provide appropriate courses for all students in grade 12.

Characteristics Common to All Three Program Models

Although the models presented here offer distinctive ways of approaching the mathematics described in the Ohio Academic Content Standards, they share several basic characteristics:

• Each demonstrates how the Standards can be implemented through a curriculum and how instruction can be organized to improve student learning;
• Each prepares students to achieve or exceed the proficiency level on the mathematics portion of the Ohio Graduation Test in grade 10 and to achieve or exceed the requirements to enter Ohio college and university mathematics courses above the remedial level by the end of the Year 3 course;
• Each clarifies where the emphases need to be in instruction and what the foci are for each course;
• Each moves students from informal experiences and intuitive understanding to levels of formal definition and logical reasoning;
• Each displays the connectedness and coherence of the mathematics studied within each course and across the courses in a sequence.

Distinctive Characteristics of the Three Models

Each model also has distinctive characteristics:

Model A. This model uses the applications of mathematics to motivate the need to master mathematical topics in algebra and geometry. By using applications to motivate the mathematics, students can become more engaged in algebraic and geometric topics, and motivated to work hard on meaningful problems. Mathematics developed in this way is intended to encourage problem solving and reasoning skills, thus preparing students well for the workplace or for further education.

Model B. This model blends the mathematics of the various content strands (algebra/number, geometry/measurement, data/statistics), weaving them together in each course and providing a sequence of courses that build on one another to form a coherent curriculum. Data topics are woven throughout the model with a focus on a data project in Year 3.

Model C. This model features a classic sequence of courses that emphasizes connections across content strands. Data analysis topics have been added to the familiar high school mathematics curriculum. Year 1 focuses on algebraic thinking and skills, augmented with data analysis. Year 2 focuses on geometric topics, both synthetic and analytic, and includes formal geometric argument. Year 3 extends the algebra topics from Year 1 and introduces traditional topics of Algebra II. Year 4 includes trigonometric functions and other topics from pre-calculus mathematics.

Each of the Models A, B, and C prepares students to take a calculus course in their first year of college. The Program Models presume that all Ohio graduates will enter postsecondary education at some time, but that not every student’s academic program will include calculus. Consequently, Models A, B, and C are followed by Model A', Model B', and Model C', respectively, which adjust the original models to provide an appropriate curriculum for students whose postsecondary program will not include the study of calculus.

A program model is a guide to assist in organizing mathematical ideas and student experiences for effective learning. However, different students learn in different ways. The amount of time, practice, and assistance students require to learn mathematics varies from student to student. These differences must be accommodated in a district’s plan for delivering the curriculum. In this section, we offer suggestions for organizing programs to accommodate student differences. We offer suggestions for three specific groups of students:

• Students entering grade 9 without the mathematical skills and understanding needed to be successful in a Year 1 course;
• Students who have completed grade 10 but not achieved or exceeded the proficiency level on the mathematics portion of the Ohio Graduation Test;
• Students with the background and abilities to be accelerated in the regular mathematics curriculum.

Preparation for the Year 1 Mathematics Course

A mathematics curriculum that reflects the Ohio Content Standards will build mathematical skills and dispositions that enable all students to understand the fundamentals of algebra. As early as pre-kindergarten, algebraic thinking activities such as finding patterns, identifying missing pieces in sequences, and acquiring informal number sense will be central parts of students’ experiences. The middle school curriculum moves students from numerical arithmetic to generalized arithmetic where symbols can represent numbers. This curriculum gives students experience with numeric, geometric, and algebraic representations of relationships. Students develop proportional reasoning skills; they investigate more complex problem settings and move from concrete experiences in mathematics to the formulation of more abstract concepts.

The Year 1 mathematics course in any secondary curriculum model is expected to be the foundation for future learning of mathematics. Formal algebra will be a focus of this course. Whether students enter the workforce directly after graduation or enter postsecondary education, success in Year 1 mathematics will be critical to their futures. There are several strategies districts should consider for students who complete grade 8 without the mathematics background needed to succeed in Year 1. These strategies assure that all students study Year 1 mathematics no later than grade 9.

Summer Sessions

During the summer prior to their Year 1 course, students could attend:

1. A focused summer course that strengthens pre-algebra methods and terminology, provides a review of basic mathematical procedures, and uses some topics of discrete mathematics to help students move from concrete thinking to generalization, or
2. A computer-based program with a teacher or coach to individualize students’ instruction and correct misunderstandings.

During the Standard School Year

In addition to summer opportunities, districts may consider the following options:

1. Provide some Year 1 mathematics classes in grade 9 that meet 8 or 10 periods a week for students who need more time to learn the mathematics in this course. Alternatively, all Year 1 mathematics classes can be taught for 8 or 10 periods a week so teachers have time to differentiate instruction and engage in extended, supervised problem solving.
2. Create a program of peer tutoring that includes training, supervision, and time for students to work with other students.
3. Create Mathematics Labs associated with specific mathematics courses (similar to labs that are linked to science courses) and to which students are assigned on a regular basis.
4. Create parent/community help teams that work under the direction of teachers and assist students with mathematics after school or during study halls.

A common feature of these strategies is that each one recognizes some students will need more time and more assistance to be successful in learning the mathematics of the Year 1 course. There are, of course, costs to each of these interventions. However, the costs of providing timely help to students is significantly less than the cost of teaching remedial courses or allowing students to enter the workforce with deficiencies in mathematics.

Reaching Proficiency Level on the OGT

Students who do not achieve or exceed the proficiency level on the mathematics portion of the Ohio Graduation Test in grade 10 can benefit from the following options:

1. Require students to attend a summer program between grades 10 and 11 in which basic concepts are reviewed and problem solving is emphasized. These students should re-take the OGT when it is offered again later in the summer.
2. Offer before school, after school, or Saturday sessions to review core mathematics topics and work with students individually; study hall periods may be used in this way for some students.
3. Develop peer-tutoring programs to help students who did not succeed on the OGT, giving peer tutors sufficient training and supervision.
4. Develop a 9-week OGT preparation course to be taken concurrently with the Year 3 mathematics course during the first grading period in grade 11. This course could also be taught during the second semester in preparation for the spring administration of the OGT. (Because the content of this short course will repeat content from earlier courses, credit for this course should not count toward the mathematics credits required for graduation.)

Students Who Are Accelerated in the Curriculum

Some students are able to move successfully through a standard mathematics curriculum at a quicker pace than the majority of students. Commitment to accelerated students must be as great as the commitment to other students to assure that they are challenged in each year of study and persist in mathematics through their senior year. Two strategies are suggested:

1. A district may designate some sections of a regular course as honors or enriched and in these sections deal with topics in greater depth, assign students more complex problems, and develop more team projects for students. Differentiating instruction in this way, rather than having a student skip a course in order to move ahead, will assure students do not miss critical material covered in each of the grade level curricula.
2. Some students may have the ability to study the Year 1 course in 8th grade if the curriculum has been modified to assure they have studied all topics of the middle school curriculum before grade 8. Because the Ohio Academic Content Standards in Mathematics identify new topics to be introduced in each of the middle grades, no mathematics course can simply be skipped. Students with the potential to be accelerated will need to be identified by the teaching staff and by readiness tests, and have their curriculum appropriately modified in the grades prior to grade 8. Students who study the Year 1 course in 8th grade should move ahead to the Year 2 course in 9th grade, continue in an enriched curriculum through grade 11, and study an advanced level mathematics course in grade 12 so they are well positioned for further study or for workplace opportunities.

Advanced Courses for Accelerated Students

The models present several options for accelerated students after they have completed the mathematics in the standard curriculum. The models include a course called Modeling and Quantitative Reasoning that provides mathematics accessible and of interest to high school students, but not always included in the high school curriculum. Another option for students who have strong backgrounds in algebra, geometry, coordinate geometry, trigonometry, and pre-calculus mathematics is a course in calculus. When a calculus course is offered for high school students, the course should be taught at the college level and students should expect it to replace a first-year calculus course in college. This can be assured by using a College Board Advanced Placement calculus course and requiring students to take the AP exam at the end of the course. In some locations, accelerated students are able to enroll in a mathematics course at an area college or take a college level course through distance education, concurrent with their high school studies. The models also prepare accelerated students to take an Advanced Placement statistics course, which can be an exciting and appropriate option.

The content in the mathematics Program Models is specified in the Ohio Academic Content Standards: Number, Number Sense and Operations; Measurement; Geometry and Spatial Sense; Patterns, Functions and Algebra; Data Analysis and Probability. The sixth standard, Mathematical Processes, is the thread that ties the five content standards together to make a meaningful and cohesive curriculum. Mathematical processes can be divided into five strands: problem solving, reasoning, communication, representation, and connections.

Authentic problem solving requires students not simply to get an answer but to develop strategies to analyze and investigate problem contexts. The National Council of Teachers of Mathematics publication, Principles and Standards for School Mathematics, states that "solving problems is not only a goal of learning mathematics but also a major means of doing so. Students should have frequent opportunities to formulate, grapple with and solve complex problems that require a significant amount of effort and should then be encouraged to reflect on their thinking." Indeed, this is how students come to understand deeply the mathematical topics in their courses.

"Reasoning involves examining patterns, making conjectures about generalizations, and evaluating those conjectures" (Ohio Academic Content Standards, K-12 Mathematics, p. 196). In mathematics, reasoning includes creating arguments using inductive and deductive techniques. Students need opportunities to make and test their conjectures, explain their reasoning, and evaluate the arguments of other students as well as their own.

Oral and written communication skills give students tools for sharing ideas and clarifying their understanding of mathematical ideas. Mathematics has its own language, and this language becomes increasingly more precise as students move through their studies. Developing skill in using this language requires students to read, write, and talk about mathematics. Understanding mathematical terminology is essential to understanding mathematical concepts.

Mathematics uses many different forms of representation to embody mathematical concepts and relationships. Some are numerical (e.g., tables); some are algebraic (e.g., expressions, equations); some are geometric (e.g., sketches, graphs); some are physical models. Students need to be comfortable using multiple representations for a single concept. This skill will help them develop problem-solving strategies and communicate mathematical ideas effectively to others. Appropriate use of technology is an essential tool for increasing students’ access to different kinds of representation in mathematics.

A coherent curriculum will help students make connections between the mathematical concepts they learned in earlier grades and the concepts they study later on. Students need to appreciate that the five content strands are not independent blocks of mathematics and that the process standard is part of learning within each content strand. Without this appreciation, students may view the content of their courses as little more than a checklist of topics. Students also need to experience the connections between mathematics and the other subjects they study. Their mathematics courses should include frequent applications drawn from the life sciences, physical sciences, social studies, and other fields. If students are to understand the importance and power of mathematics, these connections need to be explicitly discussed.

In the Program Models, these mathematical processes are developed through course design and through experiences students have when they work with rich contextual problems. Successful learning of mathematics requires that students struggle with complex problems, communicate mathematics clearly, represent mathematics accurately and in various forms, make conjectures and reason effectively, and connect mathematical concepts across the various areas of mathematics and to applications in other fields. There is no shortcut. Each of the processes must be developed in every course, in every sequence, and in every year of study.

Appropriate use of technology in the mathematics classroom is an issue that must be addressed in the development of a new curriculum. In this area, there are dual goals: (1) student proficiency with foundational skills and basic mathematical concepts using basic manual algorithms, and (2) student competence in using appropriate technology to encourage mathematical exploration and enhance understanding.

With respect to the first goal, the Program Models presume that students will enter the Year 1 course with an understanding of basic mathematical concepts and with proficiency in performing accurate pencil and paper numerical procedures. Even so, the secondary program should be designed to strengthen numerical skills and build additional skills in algebraic computation, estimation, and mental mathematics. The study of algebra, measurement, geometry, and data analysis provides useful contexts for students to continue to develop written and mental computational skills that deepen their understanding of mathematics and strengthen their abilities in problem solving.

With respect to the second goal, the Program Models presume that students will use technology as a tool in learning the mathematical concepts and working the complex problems in the secondary school curriculum. For example, technology can assist students in investigating applications of mathematics, testing mathematical conjectures, visualizing transformations of geometric shapes, and handling large data sets. Technology appropriately used can enhance students’ understanding and use of numbers and operations, as well as facilitate the learning of new concepts. Students will need to be alerted to the possibility of serious round-off error when technology is used for complex computations in real-world applications.

At this time, the Ohio Graduation Test allows students to use a state-specified scientific calculator. This calculator is primarily a computational tool, and students will need adequate time and practice using it prior to the OGT. A scientific calculator alone does not provide all the features needed to study the topics described in the Program Models. Implementing the Program Models requires decisions about the kinds of technology that students will use at different stages of their learning to assure a balanced program that results in students’ knowing when to use technology and when not to, when to use pencil and paper, and when to do mathematics in their heads. The goal, always, is to develop a program that focuses on mathematical understanding and proficiency.

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