Chapter 6

Curriculum and Program Development

            Although mathematical truths may be timeless, the content of mathematics programs and courses continues to evolve, often as a result of rapid technological advances.  This chapter focuses on the improvement of student learning in mathematics by developing, assessing, and improving mathematics courses and programs.  The intent is not to prescribe specific mathematics course content, but rather to emphasize key components of curricular design and development and encourage periodic review and revision of curricula with the goal of continuous improvement. 

Implementation Standard: Curriculum and Program Development

Mathematics departments will develop, implement, assess, and revise courses, course sequences, and programs to help students attain a higher level of quantitative literacy and achieve their academic and career goals.

The Changing Curriculum

The world has gone
business, geography,
criminal justice,
history, allied health
fields—a full range
of disciplines and job
tasks tells students
why math
requirements are not
just some abstract
school exercise.
U.S. Department of
The Toolbox Revisited:
Paths to Degree
Completion from High
School through College,
February 2006, p. xix.

Creating a responsive learning environment, assessing student learning, and improving student learning in mathematics requires a dynamic curriculum.  There are a number of factors that influence the content and organization of the mathematics curriculum.

In response to these factors, faculty need to take the lead in careful, deliberate, informed curricular revision decisions.  Research in mathematics education can provide helpful information to guide decision-making in curricular design.  The challenge is to design curricula that address the needs of as many academic paths and disciplines as possible.  To meet the diverse needs of all students mathematics departments need to collaborate with others (faculty in other disciplines, faculty in four-year institutions, business and industry representatives) to determine appropriate mathematics outcomes.  Once the outcomes are established, faculty must access the degree to which students meet those outcomes.

During the last decade, several key curricular issues have stimulated dialogue and educational research. For example, since 1989, when it was suggested that “If it does nothing else, undergraduate mathematics should help students develop function sense…,”(2) considerable research has been conducted on what it means for students to have an understanding of function.  Studies report that a well-developed understanding of function correlates closely with success in calculus, as well as facilitating the transition to advanced mathematical thinking.(3)  In addition, faculty continue to search for methods to develop a student’s understanding of the concept of variable.  Students who are able to view variables as representing quantities whose values change dynamically along a continuum have been shown to have ready access to fundamental ideas, such as rate of change and limits, and exhibit higher levels of achievement in mathematics.(4

There have also been lively discussions about what is the appropriate content of General Education Mathematics, College Algebra, and Precalculus.  For example, do students need to analyze data and construct appropriate models of real-world phenomena?  If so, then curricula need to be designed so that students understand the difference between a constant rate of change (additive, linear) and a constant growth factor (multiplicative, exponential). While learning to distinguish between the two processes, students develop fluency in pattern recognition, data analysis and proportional reasoning. (5)  These discussions, within a department, institution, or professional organization are valuable professional development opportunities for all.  Faculty and departments are encouraged to continue discussions, consult mathematics education research, and consider educational innovations when designing a curriculum.

The goal and challenge for faculty is to create courses and programs that empower students to become confident and competent problem solvers.  Courses should provide opportunities to develop the quantitative skills they will need in their academic work, in society, and in the workplace.  The dynamic nature of the curriculum requires a continuous re-examination of how, what, and when mathematics content is taught.  The focus of curricular development should be on the mathematics content that is most appropriate for student learning. 

The Beyond Crossroads Implementation Cycle can be applied to curriculum development, outlining a process for identifying goals, assessing strengths and weaknesses, and implementing curricular change as shown in Figure 3.

Curriculum Implementation Cycle

Quantitative Literacy

Take a Closer LookThe use of quantitative ideas and language is pervasive in society, news stories, scientific reports, and advertising.  Because information is often presented in a variety of representations–words, symbols, tables, or graphs–individuals need to be able to interpret, analyze, and draw conclusions about information presented in these forms. It is important that all citizens have an understanding of the magnitude of numbers, be able to compute and apply percentages, and apply the basic concepts of statistics.  In order to be informed consumers, to interpret economic and political trends, or to evaluate health risks of new drugs or treatments, individuals need to understand basic concepts of proportionality, linear and exponential growth, and elementary descriptive and inferential statistics.  While entry level jobs may not require significant quantitative reasoning, higher levels of quantitative literacy may be needed to keep open the doors to job advancement.   

LinksFor these reasons, quantitative literacy should be an outcome of all programs within the college.  While most professionals agree on the need for quantitative literacy for all students, there are variations in terminology and definition.  Quantitative literacy, quantitative reasoning, numeracy, and mathematics across the curriculum all share many common components. Definitions of quantitative literacy outline student outcomes that range from acquiring specific mathematical skills to gaining mathematical confidence.  Quantitative literacy may include the following:

Numeracy [QL] is not
the same as
mathematics, nor is
it an alternative to
mathematics. Rather,
it is an equal and
supporting partner in
helping students
learn to cope with the
quantitative demands
of modern society.
mathematics is a
discipline, numeracy
is necessarily
National Council on
Education and the
Mathematics and
Democracy: The Case for
Quantitative Literacy,
2001, p. 115.

Acknowledging that more than one definition exists, Beyond Crossroads accepts the definition of quantitative literacy presented in the basic principle for quantitative literacy in Chapter 2: an individual’s “capacity to identify, understand and engage in mathematics as well as make well-founded mathematical judgments as a constructive, concerned and reflective citizen.”(9)  Quantitative literacy includes “five different dimensions of numeracy:  practical, for immediate use in the routine tasks of life; civic, to understand major public policy issues; professional, to provide skills necessary for employment; recreational, to appreciate and understand games, sports, and lotteries; and cultural, as part of the tapestry of civilization.”(10)  While the specific definitions may vary, quantitative literacy is an important general education outcome for every college student.  Thus, quantitative literacy outcomes should be woven into every mathematics course and as many other college courses as possible.

Quantitative literacy is more about habits of mind than specific mathematical content.  Therefore, the responsibility for developing quantitative literacy, like writing across the curriculum, is shared by the entire college faculty.  However, mathematics faculty should lead the quantitative literacy movement by helping to establish a set of outcomes expected of students in each program.  There are some outcomes expected of all students.

Students in all college programs will be expected to:

Implementation recommendation: Faculty will integrate quantitative literacy outcomes into all mathematics courses and collaborate with faculty in other disciplines to integrate quantitative literacy into coursework across all disciplines.

Actions to support this recommendation

Faculty and Departmental actions:

Institutional actions:

Developmental Mathematics Courses and Programs

LinksDevelopmental mathematics courses in this document are defined to be courses below the level of the first mathematics course that earns full college credit at the institution.  For most two-year colleges, this includes mathematics courses below the level of intermediate or college Algebra. 

Developmental mathematics courses, to a greater extent than other mathematics courses, serve students with various degrees of success in previous mathematics courses.  Appropriate student placement is crucial for developing positive attitudes and maximizing success.  The goals of a solid preparation in basic mathematics and swift progress through the curriculum can be compatible.  Substantial changes to traditional developmental mathematics curricula, which were derived primarily from high school curricula, need to be implemented.  The developmental mathematics program needs to be designed to accomplish agreed-upon goals designed with input from stakeholders, using instructional strategies appropriate for diverse student learning styles and diverse teaching styles.  In order to help students who have not previously been successful in mathematics and to implement the recommendations in this report, faculty need to do more than teach the same mathematics again.

The desired student outcomes for developmental mathematics courses should be developed in cooperation with the partner disciplines.  The content for these courses also should address mathematics anxiety, develop study and workplace skills, promote basic quantitative literacy, and create active problem-solvers.  The curriculum of developmental mathematics programs should:

Every course should
incorporate activities
that will help all
students’ progress in
developing analytical,
critical reasoning,
problem-solving, and
communication skills
and acquiring habits
of the mind.
Association of
Undergraduate Programs
and Courses in the
Mathematical Sciences:
CUPM Curriculum Guide
Executive Summary, p. 1

Teaching a developmental mathematics course presents unique challenges.  Each instructor needs to provide a positive, nurturing experience for each student.  Students need to develop time management skills and study habits.  They should engage in activities to help them view mathematics as interrelated concepts of a formal system, not just as unrelated facts to be memorized.  While students need skills such as solving proportions, they also need to understand the concept of proportion and recognize its application in their lives and careers.  Students should be able to approach mathematics through contextual, concrete, and abstract situations; apply mathematical skills to solve problems; and be able to transfer their knowledge to new situations. Students should experience multi-step problems and be comfortable working in groups and doing collaborative projects.  They also should have successful experiences using technology, including calculators, spreadsheets, and other computer software, as a tool to collect, organize, and analyze data, as well as to recognize numerical and graphical patterns. 

Topics in developmental mathematics courses should be taught in depth and not as a preview of topics to come in later courses.  For many students, any given mathematics course could be their last one.  Some algebraic topics, such as factoring, radicals, and operations with rational expressions, should receive less attention, while modeling, communication, and quantitative literacy and reasoning should receive more attention.  Topics in algebra, geometry, statistics, problem solving and experience using technology should be integrated throughout developmental courses.  However, students should still be expected to perform single digit arithmetic, without the use of a calculator.   In addition to the quantitative literacy outcomes and workplace skills expected of all students, there are some global outcomes that are expected of all developmental mathematics students.

Developmental mathematics students will be expected to:

Implementation recommendation:  Mathematics departments will design and offer developmental mathematics courses that prepare students for the workplace and build mathematical skills needed for informed citizenship and success in a variety of academic and career paths. 

Actions to support this recommendation

Faculty actions:

Departmental actions:

Institutional actions:

General Education Mathematics Courses

The mathematician’s
patterns, like the
painter’s or poet’s,
must be beautiful;
the ideas, like the
colours or the words,
must fit together in a
harmonious way.
Godfrey Harold Hardy,
A Mathematician’s
1941, p. 25.

For many students, a general education course such as logic, statistics, liberal arts mathematics, finite mathematics, or discrete mathematics may be the only mathematics course required for graduation.  A general education mathematics course may not be a prerequisite to any subsequent course, but a prelude to the student’s experiences as a productive employee and informed, quantitatively literate citizen.  As a result, it is critical that general education mathematics courses exemplify the principles and standards of this document.   A general education mathematics course may mold students’ opinions of mathematics for the rest of their lives.  These courses should pay particular attention to influencing positively student attitudes toward mathematics.

General education courses should include the important general education outcomes of writing, critical thinking, and quantitative literacy. While the course goals and student outcomes may vary from program to program, all general education courses in mathematics should require students to synthesize, make connections, and use basic mathematical knowledge to solve real-world problems.  This synthesis may occur in a variety of mathematical contexts through exploration, discovery, and problem solving.  A course may be designed to focus on mathematics as a language used to describe relationships and patterns.  Another course might examine how one quantity varies with another (especially in linear, quadratic, and exponential relationships using real data) through a study of functions, geometric patterns, or statistics.  Many kinds of investigations, such as using real data, integrating algebraic or geometric structures, or studying statistics, could be used as the core of a general education mathematics course. 

In addition to the quantitative literacy outcomes expected of all students, the following are expectations of general education students.

General education mathematics students will be expected to:

Implementation recommendation:Mathematics departments will design and offer general education mathematics courses for students who take a single college-level mathematics course in their college degree program.

Actions to support this recommendation

Faculty actions:

Technical and Career Courses and Programs

Mathematics courses in technical programs include topics from arithmetic through calculus. In addition to providing necessary mathematical skills, these mathematics courses should build students’ quantitative literacy, develop their ability to think and communicate mathematically, connect mathematics to real-world situations, and develop problem-solving skills for working alone, as well as for working in teams.(11)

The application of
mathematics to
technical problems is
fundamental to all
technical education
National Science
Gaining the Competitive
Edge: Critical Issues in
Science and Engineering
Technician Education,
July 1993, p. 19.

Colleges offering technical and career programs have the responsibility of determining the needs of their local businesses and industries and preparing students to meet these needs.  Employers value thinking skills, decision-making, creativity, problem-solving, visualizing and reasoning spatially, and knowing how to learn–as well as personal qualities such as responsibility, self-management, and team skills.  Employers need college graduates with reading, writing, speaking, and listening skills, as well as knowledge of basic mathematical concepts.(12)  Designing technical courses with mathematics content for specific fields, with flexibility to be used by more than one major, is another challenge.  For example, emerging technologies, such as those in the biotechnological fields, tend to require fewer algebraic skills, and more data analysis, modeling, statistics, and discrete mathematics.

Technical mathematics courses and programs should be developed in collaboration with faculty from other disciplines and business and industry representatives to identify and address the mathematics content needs of specific program employers.  Content in two-year technical mathematics courses should be selected because of its application to a specific technical field and the needs of specific employers. It should be at a level equivalent to mathematics courses that transfer to four-year institutions.  The direction of curriculum change in technical and career programs is to emphasize workplace skills within the classroom context and to develop flexible thinkers who can work collaboratively to solve new problems. In addition to quantitative literacy outcomes, technical and career students have unique expectations from their respective programs.

Technical and career students will be expected to:

Implementation recommendation: Technical and career mathematics courses will incorporate workplace skills and address the unique needs of the programs they serve, while providing students with quantitative literacy and an appropriate mathematical background.

Actions to support this recommendation

Faculty actions:

Departmental and Institutional actions:

Teacher Preparation Courses and Programs

Two-year colleges are
essential partners in
the mathematical
education of
teachers. A large
number of future
teachers begin their
postsecondary study
in two-year colleges.
In particular,
elementary teachers
take a significant
portion of their
college mathematics
in two-year colleges.
Conference Board of
the Mathematical
The Mathematical
Education of Teachers
2001, p. 9

Many two-year colleges have the preparation of teachers as part of their mission.(13)  Many future teachers take most, if not all, of their college-level science and mathematics courses at two-year colleges.  Two-year colleges enroll a large proportion of the nation’s underrepresented groups and can assume a key role in recruiting minorities to the teaching profession.  Positioned between K-12 schools and four-year colleges, two-year colleges can play an important role in promoting collaboration in a variety of issues, such as articulation and transfer issues, curriculum development, financial support for students, use of technology, program requirements, statewide teacher licensure requirements, and professional assessments.(14)  Because of these factors, two-year colleges play a pivotal role in the recruitment and preparation of teachers. 

To be effective teachers of mathematics, “Prospective teachers need mathematics courses that develop a deep understanding of the mathematics they will teach.”(15)  Courses for prospective teachers should develop the habits of mind of mathematical thinkers and demonstrate flexible, interactive teaching styles.  They should demonstrate the role of technology to explore ideas and promote concept development.  They also should present the fundamental ideas of school mathematics and be taught by mathematicians who have a serious interest in teacher education.  All mathematics courses taken by future teachers of mathematics should be standards-based, use strategies that have demonstrated success for students, and incorporate the NCTM’s Principles and Standards for School Mathematics.(16

Some of the courses taught at two-year colleges for prospective teachers of mathematics are addressed in the report The Mathematical Education of Teachers. (17) This report includes the following recommendations to mathematics departments regarding both the content and delivery of the curriculum.

Mathematics courses must emphasize the connections within mathematics, especially connections with the mathematics that is already familiar to the students.  The courses also should illustrate the connection between mathematics and other disciplines.  Mathematics instruction for future teachers should model informed teaching practices.  These practices include setting high expectations for all students, integrating concepts, actively engaging students in the learning process, emphasizing problem solving and reasoning mathematically, and expecting students to reflect on their learning and to communicate mathematically. 

Prospective teachers should have intensive courses in effective teaching methods in mathematics.(18)  Included in these courses should be a discussion of interactions between the learner and the instructor, among learners, and between learners and the content they are trying to master.  Whole group, small group, and individual work should be utilized. Instructors should integrate assessment into instruction and use multiple assessment techniques. 

Teachers tend to teach the way they were taught and many school children are concrete learners.  Therefore, future teachers–especially future elementary teachers–need pre-service experience using manipulatives as a tool to teach mathematics.  Manipulatives help develop conceptual understanding and assist students in discovering underlying procedures.  Future teachers need experience using the same manipulatives they will encounter in preK-12 classrooms.  They also need to practice with using the Internet, spreadsheets, mathematics applets, and mathematics software, as tools to teach mathematics.(19)

In the first two years of college, prospective teachers need supervised experiences in classrooms to observe and practice what they are learning in their mathematics courses.  These early experiences help students determine whether teaching is the right career choice for them.  Fieldwork experiences prior to student teaching afford students the opportunity to begin to become reflective practitioners and evolve in their understanding of educational practices and issues as they analyze, synthesize, and summarize their experiences.  Mathematics departments should coordinate with other disciplines to afford prospective teachers the opportunity to participate in early fieldwork experiences.

In addition to providing the first two years of a college-level teacher preparation program, many two-year colleges also offer a variety of other education-related programs.  Examples include degree programs in paraprofessional education and early childhood development, professional development workshops and licensure renewal courses for teachers, and teacher licensure programs for professionals holding bachelor’s degrees who desire to switch their careers.  In some states, colleges formerly considered two-year colleges are now granting bachelor’s degrees in high-need disciplines, such as mathematics or science teaching and health care. Recommendations about the mathematical preparation of future teachers also apply to those enrolled in other education-related courses and programs at two-year colleges. The expectations of future teachers extend beyond the quantitative literacy outcomes and workplace skills outlined earlier.

Students in teacher preparation programs will be expected to:

Implementation recommendation: Mathematics departments will create a mathematics program with courses that provide future teachers of mathematics with content knowledge that is beyond and deeper than the mathematics they will be teaching and that exemplifies successful pedagogical methods.

Actions to support this recommendation

Faculty actions:

Departmental and Institutional actions:

Mathematics-Intensive Courses and Programs

In fall 1972,
President Nixon
announced that the
rate of increase of
inflation was
decreasing. This was
the first time a sitting
president used the
third derivative to
advance his case for
Hugo Rossi,
Notices of the American
Mathematical Society
October 1996, p. 1108.

Mathematics-intensive programs serve students who will major in mathematics or mathematics-dependent fields, including secondary mathematics education, business, computer science, engineering, and science.  The demand for a mathematically educated workforce has grown, but the number of students majoring in mathematics has declined.  It is essential that students, especially those from underrepresented groups, be provided with a supportive learning environment that encourages their enrollment and retention in mathematics-intensive programs.

A solid understanding of functions and proof is a critical component of students’ mathematical foundation. Each student needs to develop an ability to work at an abstract level.  Students should use rich applications from various disciplines to deepen their appreciation and understanding of the power of mathematics to model real-world phenomena.  These applications can be used to promote active and collaborative learning, practice sustained effort, examine a variety of problem solving techniques, and communicate mathematically by giving written or oral reports. Mathematics-intensive programs should increase students’ understanding of the mathematics topics studied in calculus.  The expectations of mathematics-intensive students extend well beyond the quantitative literacy outcomes expected of all students.

Mathematics-intensive students will be expected to:

Implementation recommendation: A mathematics-intensive program will promote students’ appreciation and zeal for mathematics and develop the skills, concepts, and problem solving and communication strategies necessary for success in their courses and careers.

Actions to support this recommendation

Faculty actions:


Mathematics courses and programs in the first two years of college need to develop students’ quantitative and workplace skills and actively engage them in the mathematics they will encounter outside the classroom.  Faculty may need to teach content that is different from what they were taught, teach more than they were taught, and teach differently than the way they were taught.  Students should understand some of the big ideas of mathematics through a curriculum, a variety of problem-solving strategies, and significant projects that examine selected topics in depth.  Students should have opportunities to demonstrate their mathematical knowledge, as well as their creativity.  When designing mathematics curricula, faculty and departments should consider the needs of each student, as well as the mathematical requirements of other courses and the workplace, and employ multiple approaches in instructional strategies.  In addition, each course and program should be reviewed periodically for continuous improvement. The entire mathematics community needs to embrace continuous improvement as the process that can increase student learning. 



Implementing the Standards

Curriculum and Program Development

Mathematics departments will develop, implement, assess, and revise courses, course sequences, and programs to help students attain a higher level of quantitative literacy and achieve their academic and career goals.

At a standards-based institution, the faculty:

  • integrate quantitative literacy outcomes into all mathematics courses and into coursework across all disciplines.
  • collaborate with partner disciplines and business and industry to establish desired student outcomes.
  • create courses and programs based on desired student outcomes.
  • lead the periodic review and revision of an up-to-date curriculum.
  • use technology throughout the curriculum to examine concepts from multiple perspectives, and to explore, discover, and develop technology skills needed in the workplace.
  • teach topics in developmental courses in depth (rather than a quick review of high school topics) and promote classroom and workplace skills.
  • emphasize for general education students the connections among some of the big ideas of mathematics.
  • collect, analyze, and model data from realistic applications in technical and career and other mathematics courses.
  • promote the zeal for mathematics in mathematics-intensive students by examining big ideas in depth and using multiple approaches or representations to reveal the connections among these ideas and with topics outside of mathematics.
  • expect all students to use the language and symbolism of mathematics to communicate effectively with other students and co-workers.

At a standards-based institution, the mathematics department and the institution:

  • promote quantitative literacy outcomes across the curriculum and in general education courses.
  • provide the facilities, support, and professional development for faculty to create, revise, and teach the curriculum.
  • place teacher preparation as a high priority in the department and reflect this priority in hiring, class assignments, and professional development.
  • continually review and revise curricula with input from two-year and four-year college faculty to maximize course transferability and access to additional educational opportunities.
  • support actions to increase the number of students choosing mathematics-intensive or teacher preparation programs.