# Crossroads Chapter 2 Standards for Introductory College Mathematics - 2023

**CHAPTER 2: **Standards for Introductory College Mathematics

*These standards provide a new vision for introductory college mathematics - a vision whereby students develop intellectually by learning central mathematical concepts in settings that employ a rich variety of instructional strategies.*

Mathematics and its applications should permeate the undergraduate curriculum. Mathematics programs must demonstrate connections both among topics within mathematics and between mathematics and other disciplines. Introductory college mathematics should link students' previous mathematical experiences with the mathematics necessary to be successful in careers, to be productive citizens, and to pursue lifelong learning.

Adult students entering introductory college mathematics programs today bring a rich diversity of experiences. This diversity challenges educators to define clear goals and standards, develop effective instructional strategies, and present mathematics in appropriate contexts. Institutions, departments, and individual faculty must take active roles in addressing the needs of diverse students, in providing a supportive environment, and in improving curricular and instructional strategies. The standards presented in this chapter unite many different mathematical experiences and guide the development of a multidimensional mathematics program.

The standards are based on research evidence and the best judgment of the educators who contributed to this document. They provide goals for introductory college mathematics programs and guidelines for selecting content and instructional strategies for accomplishing these goals. Given the diversity of students and institutions, it is expected that the methods used to implement the standards will vary across higher education and even within institutions.

**FRAMEWORK FOR MATHEMATICS STANDARDS**

The standards presented in this document are consistent with frameworks presented in other mathematics reform initiatives and are intended to affect every aspect of introductory college mathematics. The standards are in three categories:

**Standards for****Intellectual Development**address desired modes of student thinking and represent goals for student outcomes.**Standards for****Content**provide guidelines for the selection of content that will be taught throughout introductory college mathematics.**Standards for Pedagogy**recommend the use of instructional strategies that provide for student activity and interaction and for student constructed knowledge.

This framework for mathematics instruction will enable *all *students to widen their views of the nature and value of mathematics and to become more productive citizens.

**STANDARDS FOR INTELLECTUAL DEVELOPMENT**

At the conclusion of the first two years of their college studies, all students should have progressed in their development of certain intellectual abilities and of other competencies and knowledge. Introductory college courses across disciplines should be designed to broaden an existing educational foundation and allow students to appreciate mathematics, statistics, and data science as powerful reasoning and general problem-solving tools. AMATYC’s Standards for Intellectual Development include the areas of problem solving, modeling, reasoning, connecting with other disciplines, communicating, using technology, developing mathematical prowess, and linking multiple representations.

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**Standard I-1: Problem Solving**

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**Students will engage in relevant, authentic problem solving and mathematical and statistical thinking.**

Students will use problem-solving strategies that require persistence, analysis of assumptions, intellectual risk taking and application of appropriate procedures. These strategies should include posing questions; organizing information; constructing visual representations; solving similar, simpler problems; analyzing situations through trial and error, graphing, and modeling; and drawing conclusions by translating, illustrating, and verifying results. The students should be able to communicate and interpret their results.

Emphasizing problem solving will make mathematics more meaningful to students. The problems used should be relevant to the needs and interests of the students in the class. Such problems provide a context as well as a purpose for learning new skills, concepts, and theories.

**Standard I-2: Modeling **

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**Students will learn mathematics and statistics through modeling real-world situations. **

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Students will participate in the mathematical and statistical modeling of situations from the world around them and use the models to make predictions and informed decisions. Swetz (1991) describes the mathematical modeling process as "(1) identifying the problem, including the conditions and constraints under which it exists; (2) interpreting the problem mathematically; (3) employing the theories and ** **

**Standard I-3: Reasoning **

**Students will expand their mathematical and statistical reasoning skills as they develop convincing mathematical, statistical, and data-based arguments. **

Students will regularly apply inductive and deductive reasoning techniques to build convincing mathematical, statistical, and/or data-based arguments. They will develop conjectures based on previous knowledge, data, and intuition and test these conjectures by using logic and deductive and inductive proof, by framing examples and counterexamples, and by probabilistic and statistical reasoning. They will then draw appropriate conclusions and communicate their argument convincingly. In addition, students will judge the validity of mathematical, statistical, and/or data-based arguments using the same reasoning skills.

**Standard I-4: Connecting with Other Disciplines **

**Students will develop the view that mathematics, statistics, and data science are growing disciplines, are interrelated with human culture, and understand their connections to other disciplines. **

If students are to gain a sense that mathematics, statistics, and data science are growing disciplines, course content must include current and relatable topics such as algorithms needed for computer-based solution processes, the use of probability in understanding chance and randomization, modern approaches to statistical inference and data visualization, and the applications of non-Euclidean geometries.

These topics lend themselves to discussions of who developed the ideas, when they were developed, and what kind of human endeavors motivated their development, which reinforces recognition of math in all parts of life and cultures. Students should develop an appreciation of how mathematics and statistics provide a language for the sciences; play a role in art, music, and literature; are applied by social scientists and practitioners in health care fields; are used in business and manufacturing; and have impacted history.

**Standard I-5: Communicating**

**Students will develop the ability to read, write, listen to, and speak the languages of mathematics, statistics, and data science. **

Students will develop the skills necessary to communicate ideas and procedures, and results using appropriate mathematical and statistical vocabulary and notation. Students will develop the ability to communicate the results of analyses through appropriate models and visualizations. Furthermore, mathematics, statistics, and data science faculty will adopt instructional strategies that develop both oral and written communication skills within a context of authentic applications relevant to a diverse student population. As students learn to speak and write about mathematics, statistics, and data science, they develop acumen and become better prepared to use this knowledge and these skills beyond the classroom.

**Standard I-6: Using Technology**

**Students will use appropriate technology to enhance their thinking and conceptual understanding and to solve problems. **

Students will develop an ability to use technology to enhance their study of mathematics, statistics, and data science. Current technology can be used to aid in the understanding, exploration, and visualization of concepts and the analysis of data. Students can use technology to test conjectures, explore ideas, and verify that theorems are true in specific instances. They should also embrace technology as a tool to aid in the solution of authentic problems and to validate those solutions. Students should be able to judge the reasonableness and accuracy of the results generated by technology.

**Standard I-7: Developing Mathematical Prowess **

**Students will engage in rich experiences in the study of mathematics, statistics, data science, and related fields that encourage independent, nontrivial exploration and will develop and reinforce tenacity and confidence in their abilities and inspire them to further their studies in these fields. **

Students will develop self-confidence and persistence while engaging with mathematics, statistics, and data science problem-solving. These problems will not always have unique solutions but will provide experiences that develop the ability to conduct independent explorations. At the same time, they will learn to transfer problem-solving strategies to a variety of contexts (Druckman & Bjork, 1994) and appreciate mathematics, statistics, and data science as disciplines. They will visualize themselves using mathematics and statistics effectively in their professional work and everyday lives. They will develop an awareness of careers in mathematics and related disciplines.

**Standard I-8: Linking Multiple Representations **

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

Students will explore complex problems, using multiple approaches, and explain their solutions in both oral and written form. Students will be motivated to go beyond the mastery of basic operations, statistical algorithms, or algebraic manipulations to a real understanding of how to use mathematics and statistics, the meaning of the answers, and how to interpret them.

**STANDARDS FOR CONTENT**

AMATYC takes the position that to truly understand mathematics and statistics one must know it conceptually, contextually, and procedurally and know that problem solving is the heart of doing mathematics. The successful problem solver can view the world from a mathematical perspective (Schoenfeld, 1992).

Students develop the ability to solve meaningful problems through in-depth study of mathematics and statistics topics that build on their prior knowledge and experiences. When presented in the context of relevant applications, abstract topics grow naturally out of the need to describe or represent the patterns that emerge. In general, the meaning, use, and communication of mathematical and statistical ideas must be emphasized. Attention to rote memorization and manipulation must decrease.

AMATYC’s Standards for Content elaborates on the inclusion of threads throughout the curriculum related to numeracy, symbolism and algebra, geometry and measurement, functions, discrete mathematics, statistics and probability, and deductive proof. The standards that follow are not meant to outline a set of courses. Rather, they are strands to be included in any post-secondary mathematics pathways in whatever structural form they may take. The specific themes were selected so that learners can develop the knowledge and skills needed to be discerning citizens, making data-based decisions, and evaluating mathematical and statistical arguments. Students should also be equipped to pursue more advanced study in mathematics and other disciplines.

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**Standard C-1: Numeracy**

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**Students will accurately process, interpret, and communicate numerical information. **

“Numeracy is the ability to process, interpret, and communicate numerical, quantitative, spatial, statistical, even mathematical, information, in ways that are appropriate for a variety of contexts, and that will enable a typical member of the culture or subculture to participate effectively in activities that they value.” (Evans, 2002). Students should be able to identify and perform appropriate arithmetic operations, estimate reliably, judge the reasonableness of numerical results, understand orders of magnitude, think proportionally, and make sense of data (especially large data sets) to recognize patterns. This mathematical reasoning may be enhanced through the use of technology.

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**Standard C-2: Symbolism and Algebra **

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**Students will be able to interpret algebraic symbols, translate problems into appropriate symbolic representations, and use those representations to effectively answer questions and make predictions. **

Students will move beyond concrete numerical operations and use algebraic thinking and symbols to solve problems. Students will represent mathematical situations using a combination of appropriate symbolic, graphical, and numerical methods to form conjectures about the problems. Applications of algebraic thinking include derivation of formulas, translation of realistic problems into mathematical statements, conversion between different representations, and the solution of equations by appropriate methods.

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**Standard C-3: Geometry and Measurement **

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**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. **

Geometry is the study of visual patterns. Every physical object has a shape, so every physical object is geometric. Furthermore, mathematical objects can be represented geometrically. For example, real numbers are represented on a number line, forces are represented with vectors, and statistical distributions are represented with the graphs of curves. The use of dynamic geometry software provides for efficient integration of geometric concepts throughout the curriculum, allowing students to visualize geometric representations more effectively.

Students will demonstrate their abilities to visualize, compare, and transform objects using geometric representations. Students will develop a spatial sense including the ability to draw (either by hand or with the use of technology) one-dimensional, two- dimensional, and three-dimensional shapes from different perspectives, and extend a concept, such as vectors, to higher dimensions. Their knowledge of geometry will enable them to determine dimensions, area, perimeter, and volume of common plane and solid figures. Suggested topics might include comparison of geometric objects (including congruence and similarity), graphing, prediction from graphs, measurement, and vectors.

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**Standard C-4: Function **

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**Students will demonstrate understanding of the concept of function by several means numerically, graphically, symbolically, and verbally - and incorporate it as a central theme into their use of mathematics. **

Key curricular issues continue to stimulate dialogue and educational research. Since the National Research Council recommended in 1989 that “If it does nothing else, undergraduate mathematics should help students develop function sense...” (National Research Council, 1989, p. 51), 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 (Tall, 1992). 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. (Ursini, S., & Trigueros, M., 1997, Carlson et al.,2002)

Students will know when a relation is a function. Students will use function notation and perform operations on functions. Students will interpret functional relationships between two or more variables and formulate such relationships when presented in tabular, graphical, symbolic, or verbal representations as well as convert between representations. Suggested topics include generalization about families of functions, transformations of functions, use of functions to model realistic problems, and the behavior of functions.

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**Standard C-5: Discrete Mathematics**

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**Students will be able to recognize and use discrete mathematical algorithms and develop combinatorial abilities in order to solve problems of finite character and enumerate sets without direct counting. <**

This standard provides guidance for incorporating topics from discrete mathematics courses (which may require precalculus or calculus as prerequisites) into introductory college mathematics courses. Applications in the social and behavioral sciences, business, computing, and other areas frequently do not exhibit the continuous nature commonly treated by techniques studied in introductory college mathematics pathways. Rather, these applications involve discrete objects and focus on logic and enumeration (Dossey, 1991; Hart, 1991). The standard echoes the recommendations made in the NCTM *Principals and Standards for School Mathematics *(NCTM, 2000, p. 31) and in *Reshaping College Mathematics *(Siegel, 1989); namely, the conceptual framework of discrete mathematics should be integrated throughout the introductory mathematics pathways, as appropriate, in order to improve students' problem-solving skills and prepare them for the study of higher levels of mathematics as well as for their careers. Suggested topics in discrete mathematics may include set theory, logic, graph theory, game theory,

algorithms, proofs, sequences, series, permutations, combinations, recursion, linear programming, finite graphs, voting systems, and matrices.

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**Standard C-6: Statistics and Probability **

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**Students will use data to inform decisions and understand the world around them. **

The basic concepts of statistics, data science, and probability should be integrated throughout the curriculum using relevant contexts and appropriate technology. Students should recognize and describe variability, take variability into account when making decisions, as well as make and communicate data-based arguments. Suggested topics include appropriate methods for collecting data, creating and interpreting data visualizations, sampling variability, drawing conclusions from sample data, modeling, applications of probability, and the ethical use of data.

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**Standard C-7: Deductive Proof **

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**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. **

The use of deductive proof in mathematics sets it apart as a unique area of human endeavor. Where appropriate to enhance student understanding of mathematical concepts, mathematical proofs, including indirect proofs and mathematical induction, will be introduced. Students will engage in exploratory activities that will lead them to form statements of conjecture, test them by seeking counterexamples, and identify and, in some instances, construct arguments verifying or disproving the statements. A variety of proof techniques, including the use of manipulatives, diagrams, and pictures to create a proof without words or symbols, should also be encouraged.

**STANDARDS FOR PEDAGOGY**

When planning a lesson, an instructor should start with the question "what should students do?", rather than "what should I do?” AMATYC supports the idea that learning is a social endeavor; therefore, it is important that we humanize the culture of learning mathematics, statistics, and data science (Yeh & Otis, 2019). The most impactful classrooms use learner-centered pedagogies, such as active learning, in a classroom environment that fosters a sense of community (CBMS, 2016; NCTM, 2014). Faculty must create frequent opportunities for students to develop and demonstrate conceptual, contextual, and procedural understanding of topics. This requires pedagogical practices that may include students using concrete tools to model abstract ideas, engaging in mathematical and statistical discourse, connecting different representations of the same idea, using prior knowledge to construct new knowledge, and understanding connections between the mathematics and statistics they are learning and what they already know.

Progress has been made toward the goal of more effectively teaching students to deeply understand mathematics and statistics; however, there is a need for more faculty to consistently identify and use pedagogical strategies that promote equitable student learning. AMATYC’s Standards for Pedagogy that follow recommend the use of instructional strategies that provide for student activity and student-constructed knowledge. Evidence-based strategies which can be incorporated by most teachers without requiring substantial faculty development are highlighted in these standards. Furthermore, the standards are in agreement with the instructional recommendations contained in Common Vision (Saxe et al., 2015). The standards include active learning, making mathematical connections, multiple representations and approaches, teaching with technology, experiencing mathematics and statistics, and assessment of student learning.

** Standard P-1: Active Learning **

**Faculty will facilitate active learning that promotes increased and deeper mathematical and statistical reasoning abilities in students. Widespread implementation of high-quality active learning can help reduce or eliminate achievement gaps in STEM courses and promote equity in higher education. **

The Conference Board of Mathematical Sciences (CBMS, 2016) uses the phrase “active learning to refer to classroom practices that engage students in activities, such as reading, writing, discussion, or problem solving, that promote higher-order thinking” and calls on institutions to incorporate active learning into post-secondary instruction.

Learning occurs when students construct their own knowledge through collaboration and when students are cognitively engaged with mathematics (Smith, et al, 2021). Participation in mathematical and statistical discourse, as well as writing and reading about mathematical and statistical ideas teaches students how to communicate about mathematics both orally and in writing. This creates a sense of community in the classroom and allows students to learn to work effectively to solve challenging problems. “For students from different socioeconomic, cultural, and educational backgrounds, and for students with different approaches to learning and social interaction, a supportive community of learners can be cultivated using AL techniques.” (CBMS, 2016, para. 13) “Working in groups also provided less confident or less able students with opportunities to explain, question, agree and disagree and test their thinking in a less threatening context” (Sharma, 2015).

**Standard P-2: Making Mathematical Connections **

**Faculty will actively involve students in meaningful mathematics work that connects to students’ experiences and focuses on broad mathematical and statistical themes that build connections within branches of mathematics, and with other disciplines. Students will view mathematics and statistics as relevant to their lives. Making mathematics and statistics relevant and meaningful is the collective responsibility of faculty, administrators, and producers of instructional materials. **

Traditionally, there has been a disconnect between classroom mathematics and real-world mathematics. Mathematics and statistics must not be presented as isolated sets of rules and procedures, but rather as disciplines that arose out of, and are connected to the needs of other fields. Further, students should be encouraged to make explicit connections between mathematical concepts, including those that may have been traditionally compartmentalized. Topics learned in one branch of mathematics should be explicitly aligned with topics from another, for example how principles learned in arithmetic can be generalized to principles in algebra, which can then be connected to topics in geometry.

Understanding that mathematics and statistics have relevance to their life and to the world in general improves student motivation to learn and ability to connect ideas. Students who understand the role that mathematics and statistics have played in their cultures and the contributions of their cultures to mathematics and statistics are more likely to persevere in their study of the discipline. Faculty should include aspects of mathematics history and contemporary mathematics that provide counterexamples to the pervasive Eurocentric bias found in modern mathematics. Instructional activities should provide examples of how mathematics and statistics are used in a variety of cultures, and by people of every race, ethnicity, gender identity, class, and other social groups. Additionally, instruction should be culturally relevant, culturally responsive, and culturally sustaining (Paris & Alim, 2017).

**Standard P-3 Multiple Problem-Solving Strategies **

**Faculty should help students become flexible problem solvers by allowing students to discover multiple problem-solving strategies and to identify efficient strategies. **

Flexibility in problem solving is an essential element of mathematical proficiency (CCSSI, 2012). Faculty should provide opportunities for students to discover their own problem-solving strategies and reflect on them (Star & Rittle-Johnson, 2008). Flexibility develops from exposure to multiple methods, comparison of worked examples, prompting and direct instruction, invention of a second method for a previously solved problem, and the opportunity to collaborate with peers (Newton et al., 2020). Experience with multiple problem-solving strategies helps students adaptively choose more efficient strategies based on the content or context of the problem (Rittle-Johnson & Star, 2007).

**Standard P-4 Multiple Representations of Mathematical Concepts **

**Faculty will provide opportunities for students to use, share, and make sense of multiple representations of mathematical and statistical ideas. These multiple representations may include words, equations, different algebraic notations, graphs, diagrams, models, manipulatives, and computer code. **

Mathematics and statistics are connected webs of knowledge where conceptual knowledge links individual pieces of information. “The development of this conceptual knowledge can only be done so by the construction of relationships between pieces of information” (Hiebert, 1986). “The skills that are at the focal point of conceptual learning in mathematics are the ability to identify and express the same concept in different forms of representation, to choose the most appropriate representation from among the various representations, and to be aware of the advantages and disadvantages of the representations” (İncikabı, 2017). Using multiple representations broadens and deepens the connections students make between concepts (Abell et al., 2018; Knill, 2009). This will motivate students to go beyond the mastery of basic operations to a deeper understanding of how to use mathematics and statistics, the meaning of the answers, and how to interpret them (NRC., 1989).

**Standard P-5: Teaching with Technology **

**Faculty will use appropriate technology to promote deeper student learning and will model the use of technology**.

Technology is an integral part of modern mathematics and statistics instruction. Faculty should be purposeful in their selection of technology, considering how it aids learning mathematical, statistical, and data science ideas. Pedagogy will include the use of technology to solve, model, and investigate mathematical and statistical problems and will provide students with opportunities to develop conceptual understanding. Emphasis should be placed on the use of high-quality, flexible, accessible technologies that enhance learning. The use of tools that students are likely to encounter in future work and careers, such as statistical software and web-based apps, is essential.

**Standard P-6: Experiencing Mathematics and Statistics **

**Faculty will provide learning activities beyond the scope of the classroom that promote independent thinking and challenge students to persistently pursue efforts over an extended time period. **

Faculty should seek opportunities to expand student knowledge of how mathematics and statistics are used beyond the scope of the classroom by providing learning activities, including open-ended projects and research opportunities. In addition, they should help their institutions form partnerships with area businesses and industries to develop opportunities for students to have realistic career experiences (Reich, 1993). Such activities will enable students to acquire the confidence to access and use needed technical information, and to independently form conjectures from an array of specific examples, and to draw conclusions from general principles.

**Standard P-7: Assessment of Student Learning **

**Faculty will incorporate multiple strategies for formative and summative assessments to inform future pedagogical practices and to help students recognize their current understanding. **

Formative and summative assessments are complementary tools for assessing the progression of student learning and informing instruction. Formative assessment benefits students and faculty by helping them recognize students’ current knowledge and setting goals for future understanding. Formative assessment takes place regularly during a term and is designed to be low-stakes and informative. Any activity that gives students an opportunity to engage with feedback to improve their understanding is an opportunity for formative assessment. Another goal of formative assessment is to inform teaching practices and strategies to best meet the needs of learners. Good formative assessment produces significant, and often substantial, learning gains (Black & William, 2005).

Formative assessment is most effective when the following principles are applied (Gehrtz, Brantner, & Andrews, 2022; Purcell, 2014; Yale University, n.d.).

• Regularly refer to the learning objectives and explicitly connect them to the learning activities.

• Watch and listen to students as they work to understand student thinking before intervening. Ask open-ended questions that provide opportunities for students to further describe and explain their thinking and reasoning.

• Use qualitative oral and written comments that help students recognize what they understand and what they need to do to increase understanding.

• Adapt teaching plans as a result of the formative assessment outcomes.

• Useful and timely feedback is essential for assessments to lead to learning (GAISE, 2016).

Summative assessments are for the purpose of evaluating student learning and assigning grades. It is especially important to ensure that the assessment aligns with the goals and expected outcomes of the instruction. Instructors should use multiple forms of summative assessment such as projects, portfolios, and demonstration of understanding in authentic situations. Instructors should consider the following principles when designing summative assessments (Blonder, et al.; Yale University, n.d.).

• Design clearly understood questions that align with learning objectives.

• Provide an opportunity for students to demonstrate their understanding of how the foundational concepts of the course are interrelated and can be applied beyond the course contexts.

• Provide opportunities to close the gap between current and desired performance, such as opportunities for resubmission.

• Consider matters of equity to ensure all students have opportunities to succeed. This may require flexible structure in conducting assessments. Flexible assessments, such as team quizzes, take home assignments, and projects provide more equity and inclusion in math courses.

**GUIDELINES FOR ACHIEVING THE STANDARDS**

Faculty who teach introductory college mathematics must increase the mathematical power of their students. This power increases the students' options in educational and career choices and enables them to function more effectively as citizens of a global community where the opportunities offered by science and technology must be considered in relation to human and environmental needs. In order to achieve these goals, mathematics education at the introductory level requires reform in curriculum and pedagogy.

The idea that mathematics competence is acquired through a curriculum that is carefully structured to include the necessary content at the appropriate time and the use of diverse instructional strategies is an underlying principle of this document. The following table provides guidance for change in the content and pedagogy of introductory college mathematics. When items are marked for decreased attention, that does not necessarily mean that they are to be eliminated from mathematics education. Rather, it may mean that they should receive less attention than in previous years, or that their in-depth study should be moved to more advanced courses where they may be immediately applied. On the other hand, increased emphasis must be placed on the items listed in the increased attention column in order to achieve the goals set forth in this document.

**SUMMARY**

These standards provide a new vision for introductory college mathematics-a vision whereby students develop intellectually by learning central mathematical concepts in settings that employ a rich variety of instructional strategies. To provide a more concrete illustration of these standards, the Appendix contains a set of problems that brings them to life.