MATHEMATICS PROGRAM MODELS FOR OHIO HIGH SCHOOLS
The ODE Mathematics Program Models offer six (6) different sequences of courses that take an applications,
blended, or connected approach to the high school mathematics curriculum. The ORC Pacing Guides (upper left navigation
bar) feature a schedule of topics, links to best practice lessons, teaching tips, and rich problems to engage students
in exploration, analysis, and application of big ideas in mathematics.
(Below is condensed from Ohio Department of Education draft, June 2006)
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.
Descriptions of the Mathematics Program Models
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.
Success for All Students
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
- 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.
During the summer prior to their Year 1 course, students could attend:
- 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
- A computer-based program with a teacher or coach to individualize students’ instruction and correct
During the Standard School Year
In addition to summer opportunities, districts may consider the following options:
- 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
- Create a program of peer tutoring that includes training, supervision, and time for students to work with other
- 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.
- 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:
- 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.
- 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.
- Develop peer-tutoring programs to help students who did not succeed on the OGT, giving peer tutors sufficient training
- 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:
- 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.
- 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
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
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.
Possible Course Sequences