Beyond Crossroads

Implementing Mathematics Standards in the First Two Years of College

Shaping the Vision

LinksIn 1995, the American Mathematical Association of Two-Year Colleges (AMATYC) released its first standards document, Crossroads in Mathematics (1).  Crossroads in Mathematics emphasized desired modes of student thinking and guidelines for selecting content and instructional strategies.  The purposes of AMATYC’s second standards document, Beyond Crossroads, are to renew and to extend the goals, principles, and standards set forth in Crossroads and to continue the call for their implementation.  Beyond Crossroads presents a renewed vision for mathematics courses offered in the first two years of college with an additional set of standards, called Implementation Standards, which focus on student learning and the learning environment, assessment of student learning, curriculum and program development, instruction, and professionalism. 

The ultimate goals of this document are to improve mathematics education and to encourage more students to study mathematics.
Crossroads in Mathematics: Standards for Introductory College Mathematics Before Calculus, 1995, p.1

Beyond Crossroads is intended to stimulate faculty, departments, and institutions to examine, assess, and improve every component of mathematics education in the first two years of college.  The varied challenges for full-time and adjunct faculty are fully acknowledged.  Faculty need and deserve the necessary facilities, equipment, and professional development opportunities essential for performing their teaching responsibilities.   The standards, recommendations, and action items are not intended to be a prescription for action used identically by each faculty member, department, or institution.  Rather, they are to be used as a starting point for dialogue, reflection, experimentation, evaluation, and continuous improvement.  Used in this way, this document can guide professionals toward standards-based mathematics education that promotes continuous professional growth and helps students maximize their potential in every college mathematics course.

What is a mathematics standard?  Mathematics faculty, administrators, mathematics education researchers, policymakers, politicians, and parents continue to engage in dialogue on the meaning and role of standards.  The words “educational standard” can have any of the following meanings depending on the audience and the purpose for which the standard is developed: (2)

“Nationally developed standards in mathematics, science, and technology represent a set of fundamental changes in the way these subjects have traditionally been taught…”(3)  The Standards for Intellectual Development, Content, and Pedagogy of Crossroads in Mathematics focused on contentand instruction, describing what students should know and be able to do in mathematics and outlining pedagogical principles.  To accomplish those standards, additional levels of support are needed.(4)  The Implementation Standards of Beyond Crossroads address the key elements of student expectations, facilities, human resources, materials, curriculum design, instructional delivery formats, and professional development to achieve the standards of Crossroads in Mathematics

One of the goals of Beyond Crossroads is to clarify issues, interpret, and translate research to bring standards-based mathematics instruction into practice.  Two definitions from the literature of standards-based education have been adopted: 

LinksTo accomplish this alignment, Beyond Crossroads has integrated recommendations from AMATYC position statements and related mathematics organizations: Principles and Standards for School Mathematics of the National Council of Teachers of Mathematics (NCTM)(7) and Undergraduate Programs and Courses in the Mathematics Sciences: CUPM Curriculum Guide 2004, a report of the Committee on the Undergraduate Program in Mathematics of the Mathematical Association of America (MAA).(8)  

The Implementation Standards of Beyond Crossroads are first presented in Chapter 3.  Then each of Chapters 4-8 focuses on one Implementation Standard.  Sections within those chapters address key issues for implementing standards-based mathematics education and include the following:

I advise my students
to listen carefully the
moment they decide
to take no more
mathematics courses.
They might be able to
hear the sound of
closing doors.

James Caballero,
Everybody a
CAIP Quarterly,
Fall 1989, 2(2), p. 2

The focus of the document is two-year college mathematics education and lower division mathematics education at four-year colleges and universities with characteristics similar to two-year colleges.  The primary audience for Beyond Crossroads is two-year college mathematics faculty(i) . Since faculty who teach lower division mathematics at four-year colleges and universities are faced with similar issues as two-year college faculty, they are also an important audience.  Additional audiences include college administrators, K-12 teachers, policy makers, government agencies, professional societies, publishers, and funding agencies.  Individuals, organizations, and businesses outside of education are called upon in Beyond Crossroads to collaborate with educators and each other, to improve college mathematics programs, and to respond to the needs of the mathematics community.  This document serves as a call to action for all stakeholders to work together to improve student success in mathematics courses and programs in the first two years of college, with particular emphasis on the two-year college student, faculty, and institution characteristics.

Distinctive Characteristics of Two-Year Colleges, Students, and Faculty

The distinctive characteristics of two-year colleges, students, and faculty make a compelling case for the development and implementation of distinguishing standards for mathematics in the first two years of college. In the century since its inception,(9) the two-year college has grown to offer a wide range of transfer, technical, and career-specific courses and programs to a diverse student population. 

There were more than 1,150 two-year colleges serving 10.1 million students, with 6.6 million enrolled in credit classes in the year 2005.  A two-year college was within commuting distance of nearly every person in the United States.  In the academic year 2001-2002, 53 percent of all undergraduate students in the United States were enrolled at two-year colleges.(10)  Most two-year colleges offered these courses, programs, and services:


In the academic year 2001-2002, two-year college students had these characteristics:(11)

Approximately 1.3 million students enrolled in courses in the following mathematics courses at two-year colleges.

Table 1.  Percent of Students Enrolled in Mathematics Courses
at Two-Year Colleges in 2005 (12)

Mathematics course


Developmental mathematics








Other mathematics courses*


*These include linear algebra, probability, discrete mathematics, finite mathematics, mathematics for liberal arts, mathematics for elementary school teachers, technical mathematics, and computing.

The 8,793 full-time permanent faculty teaching mathematics in two-year colleges in the year 2005 had the following characteristics: (13).

Full-time faculty generally taught more higher-level courses and adjunct faculty taught more lower-level courses. 

 Table 2:  Percent of Sections Taught by Full-Time and Adjunct Faculty in Two-Year Colleges in 2000(14)

Mathematics Course



Developmental mathematics



Technical mathematics









Nonmainstream calculus



Mainstream calculus for science majors



Advanced level
Service courses

Other mathematics courses



*These include linear algebra, probability, discrete mathematics, finite mathematics, mathematics for liberal arts, mathematics for elementary school teachers, technical mathematics, and computing.

In the year 2005, adjunct faculty taught 44 percent of all two-year college mathematics sections and had these characteristics:

The Standards of Crossroads in Mathematics (1995)

Crossroads in Mathematics outlined three sets of standards that provide the foundation for Beyond Crossroads:

Standards for Intellectual Development outline guidelines for desired modes of student thinking and goals for student outcomes.  All students should develop certain intellectual mathematical abilities as well as other competencies and knowledge.  Introductory college mathematics courses and programs should help students see mathematics as an enriching and powerful discipline. The seven Standards for Intellectual Development outlined in Crossroads are presented below, with the addition of an eighth standard.

Problem solving.  Students will engage in substantial mathematical problem solving.

Modeling.  Students will learn mathematics through modeling real-world situations.

Reasoning.  Students will expand their mathematical reasoning skills as they develop convincing mathematical arguments.

Connecting with other disciplines.  Students will view mathematics as a growing discipline, interrelated with human culture, and understand its connections to other disciplines.

Communicating.  Students will acquire the ability to read, write, listen to, and speak mathematics.

Using technology.  Students will use appropriate technology to enhance their mathematical thinking and understanding and to solve mathematical problems and judge the reasonableness of their results.

Developing mathematical power.  Students will engage in rich experiences that encourage independent, nontrivial exploration in mathematics, develop and reinforce tenacity and confidence in their abilities to use mathematics, and be inspired to pursue the study of mathematics and related disciplines.

Linking multiple representations.  Students will select, use, and translate among mathematical representations–numerical, graphical, symbolic, and verbal–to organize information and solve problems using a variety of techniques.

Standards for Content outline guidelines for selecting the content that will be taught. “Knowing mathematics” means being able to do mathematics.  Students gain the power to solve meaningful problems through in-depth study of mathematics topics.  The meaning and use of mathematical ideas should be emphasized and attention to rote manipulation de-emphasized.  Following are the seven Standards for Content outlined in Crossroads are presented below, with some revision. 

Number sense.  Students will perform arithmetic operations, as well as reason and draw conclusions from numerical information.

Symbolism and algebra.  Students will understand the use of algebraic symbolism,  be able to translate problem situations into symbolic representations, and use those representations to solve problems.

Geometry and measurement.  Students will develop a spatial and measurement sense, learn to visualize and use geometric models, recognize measurable attributes, and use and convert units of measure.

Function sense.  Students will demonstrate understanding of the concept of function–numerically, graphically, symbolically, and verbally–and incorporate this concept into their use of mathematics.

Continuous and discrete models.  Students will be able to recognize and use continuous and discrete models to solve real-world problems.

Data analysis, statistics, and probability.  Students will collect, organize, analyze, and interpret data, and use that information to make informed decisions.

Deductive proof.  Students will appreciate the deductive nature of mathematics as an identifying characteristic of the discipline, recognize the roles of definitions, axioms, and theorems, and identify and construct valid deductive arguments.

Standards for Pedagogy outline guidelines for instructional strategies in active student learning. Instructional strategies have a dramatic impact on what students learn.  Students should understand mathematics as opposed to performing memorized procedures.  Knowledge cannot be “given” to students.  Students should construct their own knowledge, and monitor and guide their own learning and thinking.  The five Standards for Pedagogy outlined in Crossroads are presented below, with some revision.

Teaching with technology.  Mathematics faculty will model the use of appropriate technology in the teaching of mathematics so that students can benefit from the opportunities technology presents as a medium of instruction.

Active and interactive learning.  Mathematics faculty will foster interactive learning through student writing, reading, speaking, and collaborative activities so that students can learn to work effectively in groups and communicate about mathematics both orally and in writing.

Making connections.  Mathematics faculty will actively involve students in meaningful mathematics problems that build upon their experiences, focus on broad mathematical themes, and build connections within branches of mathematics and between mathematics and other disciplines.

Using multiple strategies.  Mathematics faculty will use multiple instructional strategies, such as interactive lecturing, presentations, guided discovery, teaching through questioning, and collaborative learning to help students learn mathematics.

Experiencing mathematics.  Mathematics faculty will provide learning activities, including projects and apprenticeships, that promote independent thinking and require sustained effort.


The teaching and learning of mathematics in lower division mathematics courses and assessing what is successful learning presents ongoing challenges to students, faculty, departments, and institutions. Professionals continue to search for strategies to address a variety of issues facing mathematics education such as:

The environment for learning and teaching mathematics in higher education continues to change.  Mathematics in the first two years of college holds the promise of opening paths to mathematical power and adventure for a segment of the student population whose opportunities might otherwise be limited.  Mathematics education at this level plays such a critical role in fulfilling people’s careers in a global, technological society, that its improvement is essential not only to each individual, but also to the nation.  Beyond Crossroads challenges all faculty, departments, and institutions to adopt a philosophy that includes informed decision-making and continuous improvement in order to implement the principles and standards presented in the following chapters.