Understanding how students learn mathematics and knowing which instructional methods are likely to be successful should inform instructional practice. Effective mathematics instruction requires a variety of resources, materials, technology, and delivery formats that take into account students’ different learning styles and instructors’ different teaching styles. Every teaching activity should promote active learning and be guided by informed decisionmaking.
Implementation Standard:InstructionMathematics faculty will use a variety of teaching strategies that reflect the results of research to enhance student learning. 
“Teaching style” refers to an instructor’s contentindependent, persistent qualities, attitudes and traits. It is directly linked to the instructor’s educational philosophy and a subset of the instructor’s life philosophy.(1) Teaching styles may be studentcentered or teachercentered or any combination of the two. In a studentcentered style, the instructor expects the student to take responsibility for learning and the student trusts the instructor will help. Studentcentered models involving a constructivist approach and active learning have been shown to be successful for students with nontraditional learning styles.(2) In a teachercentered style, the instructor organizes the transmission of content knowledge to the learner, acts as a performance assessor, and provides opportunities for the learner to practice independently. Understanding where an instructor fits in the continuum of styles and using that information to make informed decisions about instruction is one of the steps in becoming a professional educator, rather than simply a teacher. Regardless of which teaching style is preferred, using multiple instructional strategies is recommended to address as many individual learning styles (see Chapter 4).
Table 6. StudentCentered and TeacherCentered Teaching Styles(3)
StudentCentered 
TeacherCentered 
Knowledge is constructed by students gathering and synthesizing information 
Knowledge is transmitted from instructor to students 
Emphasis is on using knowledge to address problems students will experience outside of the classroom 
Emphasis is on the acquisition of knowledge, sometimes outside the context in which it will be used 
The instructor’s role is coach, facilitator, and mentor; students and instructor assess learning performance together 
The instructor’s role is primarily as an informationgiver and performance assessor 
Assessment is used to diagnose learning problems, in addition to evaluating learning outcomes 
Assessment is used to evaluate learning outcomes 
Teaching and assessment of learning are intertwined 
Teaching and assessment of learning are separate 
The learning culture is cooperative and collaborative 
The learning culture is competitive and individualistic 
Instructors and students learn together 
Students are viewed as learners 
The diverse needs of students in any mathematics course call for diverse approaches to instruction. While knowledge of content is essential in teaching any discipline, effective teaching is the result of integrating content and pedagogical knowledge.(4) Effective instructors venture outside their comfort zone, take risks, grow in their teaching, and model the behavior they expect from their students.(5) The instructionalmethods chosen can encourage or inhibit students’ ability to demonstrate their mathematical knowledge and take ownership for creating and understanding mathematical ideas.(6) Certain teacher practices such as providing students with frequent opportunities to experience problem solving in a variety of contexts, allowing adequate time for solving problems, devoting time specifically for planning, monitoring, and reflecting on progress and the answer, have been shown to help students improve problemsolving skills. “A variety of instructional strategies are necessary to respond to as many different learning and teaching styles as possible.”(7)
...students who can think mathematically and reason through problems are better able to face the challenges of careers in other disciplines— including those in nonscientific areas. 
Mathematical Association of America’s Curriculum Renewal Across the First Two Years Subcommittee (CRAFTY), The Curriculum Foundations Project, Voices of Partner Disciplines, 2004, p. 1 
Mathematics instruction may be improved through sensitivity to learning styles. However, matching teaching style to learning style has limitations and does not necessarily guarantee greater student achievement.(8) One metastudy involving children concluded that there was little effect on learning when a teaching strategy was chosen specifically to respond to a student’s particular learning style.(9) In other research involving college students, matching students’ learning styles with a variety of instructional strategies showed some success, particularly with new students and poorly prepared students, where most attrition occurs.(10) To become more aware of the kinds of methods an instructor is selecting, the instructor can code lesson plans by placing a “V” next to visual activities, an “R” next to reflective activities, etc. “If the coding system is used on a regular basis, it becomes very natural to think in terms of being inclusive, or providing the setting and the activities by which all learners can find some portion of the class that particularly appeals to them.”(11) In selecting instructional strategies, the goal should be to find the best instructional strategy for the particular content being taught(12) and for the particular level of cognitive development of the students in the class.
Implementation recommendation: Faculty will recognize their own teaching style(s), reflect on the implications of their style on their students’ learning styles, and use that knowledge and other research to make informed decisions about the selection of multiple instructional designs and their classroom management. 
Actions to support this recommendation
Faculty actions:
Many college students today have grown up with technology and as a result, they process information differently than students who did not. This characteristic has implications for mathematics classes, in general, not just courses using calculators or computers. When planning classroom activities, instructors need to acknowledge that their students:(13)
For today's students, learning is participatory  knowing depends on practice and participation.
New findings in the science of learning suggest that learning requires some sort of cognitive restructuring of material by the learner. “Transfer can be improved by helping students become more aware of themselves as learners who actively monitor their learning strategies and resources and assess their readiness for particular tests and performances."(14) Therefore, instructors need to draw out and work with preexisting understandings students bring with them.
If students in mathematics classes are to learn mathematics with understanding...then it is important to examine the roles of the teacher and the knowledge that underlies the teacher’s enactments of those roles. 
National Research Council, How People Learn: Brain, Mind, Experience, and School. 1999, p. 152 
Collaborative and cooperative learning are examples of strategies to encourage active student learning. Collaborative learning is defined strictly as an unstructured process in which participants define problems, develop procedures, and produce socially constructed knowledge. Collaborative learning develops social skills, promotes studentstudent and studentfaculty interactions and builds selfesteem in students as it develops mathematics skills and knowledge.(15) A shy student, when placed in a small group, still benefits from collaborative learning by observing strategies used by his/her peers. One of the most effective means of restructuring is by explaining the material to someone else.(16) Cooperative learning is a structured, systematic instructional strategy in which groups work toward a common goal.(17) In practice, most group learning activities are a mix of cooperative and collaborative learning. The goal of both collaborative and cooperative learning is for students to be actively involved in learning activities that lead to a rich understanding of the course content.
Interactive lecturing, question posing, and inquirybased strategies have also been shown to increase students’ active learning, interest, and curiosity.(18) Designing questions and problems that are challenging yet within student’s cognitive development and giving students time to respond are critically important for these instructional methods to be effective. Discoverybased learning utilizing carefully designed questions and activities can lead students to connect new knowledge to previous knowledge. Additional questions may be necessary to facilitate knowledge construction for some students. Allowing a learner to grapple with the logic behind a rule and then apply it may be frustrating, but develops understanding of concepts. (19)
The appropriate use of writing in mathematics courses is another strategy that can play an important role in the process of internalizing mathematical procedures, understanding the relationships of mathematical concepts, and synthesizing different mathematical components into a coherent schema.(20) Writing “is an essential activity to create order from chaos, sense from nonsense, meaning from confusion: as such it is the heart of creative learning in both the arts and sciences.”(21)
Active learning occurs in many formats such as collaborative learning, discoverybased learning, interactive lecturing and question posing, and writing. Whichever format is chosen, the goal of the activity should be to enhance conceptual understanding.
Table 7. Instructional Strategies that Promote Active Learning

Strategies for Faculty: 
Collaborative/ 

Discoverybased Learning 

Interactive Lecturing and QuestionPosing 

Writing 

Implementation recommendation: Faculty will design and implement instructional activities that actively engage students in the learning of mathematics. 
Actions to support this recommendation
Faculty actions:
Departmental/Institutional actions:
Developments and advances in technology–hardware and software–have had a tremendous impact on our lives. The infusion of technology into education presents interesting opportunities for teaching and learning, especially in mathematics. Technology changes not only how mathematics is taught, but also when and what mathematics is taught.
The definition and use of technology in mathematics education is constantly evolving. Technology may refer to the use of graphing calculators, student response systems, online laboratories, simulations and visualizations, mathematical software, spreadsheets, multimedia, computers or the Internet, and other innovations yet to be discovered. Technology can be used to learn mathematics, to do mathematics, and to communicate mathematical information and ideas. The Internet hosts a wealth of mathematical materials that are easily accessible through the use of search engines, creating additional avenues to enhance teaching and facilitate learning. Outside of class, students and faculty can pose problems and offer solutions through email, chat rooms, or websites. Technology provides opportunities for educators to develop and nurture learning communities, embrace collaboration, provide communitybased learning, and address diverse learning styles of students and teaching styles of teachers.(22)
Faculty using technology should answer the question: How can the use of technology facilitate learning that is durable, has substance, is engaging to students, and provides mathematical insights through a high level of understanding of the mathematics being taught? They should expose students to a variety of technology options appropriate for solving different types of problems and guide them in making appropriate choices about the use of technology for problem solving.
Technology enhances the learning of mathematics when used appropriately as a tool, to achieve the following:
In addition, technology helps students document the validity of their mathematical/critical thinking process, facilitating and enriching the learning processes and the development of problemsolving skills.
The use of technology should be guided by consideration of what mathematics is to be learned, the ways students might learn it, the research related to successful practices, and the standards and recommendations recommended by professional organizations in education. Technology can be used by mathematics educators to enhance conceptual understanding through a comparison of verbal, numerical, symbolic, and graphical representations of the same problem. Students can use technology to search for patterns in data, while allowing the technology to perform routine and repeated calculations. The use of technology should not be used as a substitute for an understanding of and mastery of basic mathematical skills. Technology should be used to enhance conceptual understanding, while simultaneously improving performance in basic skills.
We are educating a generation of “natives,” who have grown up using sophisticated communications and information technologies. This fact alone has helped to transform the way technology is being used and integrated into classroom instruction. 
National Science Foundation, America’s Pressing Challenge—Building a Stronger Foundation, 2006, p. 6 
The integration of appropriately used technology can enhance student understanding of mathematics through pattern recognition, connections, and dynamic visualizations. Electronic teaching activities can attract attention to the mathematics to be learned and promote the use of multiple methods. Learning can be enhanced with electronic questioning that engages students with technology in small groups and facilitates skills development through guideddiscovery exercise sets. Using electronic devices for communication, all students can answer mathematics questions posed in class and instructors can have an instantaneous record of the answers given by each student. This immediate understanding of what students know, and don’t know, can direct the action of the instructor in the teaching session.(24) This cyclic assessment of student learning is an illustration of the Beyond Crossroads Implementation Cycle in action. Faculty establish goals for the use of technology in the classroom, establish assessments to measure the activity, and collect and analyze the data to revise and improve the activity.
Students will be expected to use technology to:
Implementation recommendation: Faculty will integrate technology appropriately into their teaching to enhance students’ understanding of mathematical concepts and skills. 
Actions to support this recommendation
Faculty actions:
Departmental/Institutional actions:
Distance learning, or learning at a distance, involves alternative delivery formats where the instructor and the student are separated either by time, location, or both. Distance learning courses may be delivered synchronously (in realtime), asynchronously (in delayed time), or a combination of both (hybrid) through written correspondence, text, graphics, various digital media, interactive television, videoconferencing or online. Instruction can involve a variety of distance learning delivery formats:
Although colleges and universities have been offering online instruction since the early 1980s, the World Wide Web has provided them with new means of reaching out not only to their traditional service areas but well beyond... 
M. Simonson, S. Smaldina, M. Albright, and S. Zvacek, Teaching and Teaching at a Distance, 2003, p. 233. 
Faculty across the nation have developed and implemented distance learning courses and programs in an effort to provide alternative delivery formats, flexibility in course scheduling, and accessibility for students. Yearly reports from the National Center for Educational Statistics show an increase in the number of courses offering some degree of technologybased delivery.(25) Education at a distance allows students with work and personal commitments to take courses at their convenience and participate in a dynamic and interactive process for student learning.(26)
The extraordinary growth of technologymediated mathematics courses in higher education has created new challenges. The pressure to expand the number of courses taught at a distance must be balanced with sound strategies for assessment to assure quality. Students and faculty face new challenges with regard to communicating mathematics through these different media. Faculty face new challenges as they attempt to reach the same goals for problem solving, critical thinking, collaboration, communication, content, and assessment as they have in a seated class. These challenges require that professional organizations provide faculty with opportunities to share experiences and learn from one another through conferences, workshops, and by participating in electronic discussions sponsored by the organizations.
Mathematics distance education programs must be carefully planned so they effectively deliver distance education and achieve the same goals and objectives and student learning outcomes as the same course taught in a more traditional classroom. Developing interactive distance learning tools to promote active learning in mathematics requires appropriate infrastructure, professional development, and resources. Policy and accrediting agencies have outlined benchmarks and factors such as institutional context and commitment, curriculum and course development, teaching and learning, faculty support, student support, and evaluation and assessment to ensure excellence in distance education and reflect quality.(27)
The capacity of the college media and computing services to support a distance learning program is an important consideration. Access and equity must be assured for all students. Student outcomes, expectations, and responsibilities must be clearly outlined and communicated. Orientation and advising, specifically related to distance learning, must be provided to students. Students must be aware of the personal attributes required to be successful in courses that have an alternative delivery format. Faculty and departments should develop strategies that help students make wise decisions about which delivery format is most likely to result in success. In addition, students must have easy access to the hardware and software required in the course. Faculty should consider requiring students to use the platform on which the course will be delivered as a prerequisite to registration. Essential components of a distance learning course include the following:(28)
The Implementation Cycle of Beyond Crossroads can be helpful in defining goals and objectives, assessing effectiveness, and making improvements in distance learning courses on a continuing basis.
Students will be expected to do the following :
Implementation recommendation: Faculty will utilize effective instructional design practices when developing and implementing distance learning courses in mathematics. 
Actions to support this recommendation
Faculty actions:
Departmental/Institutional actions:
The teaching of mathematics should be guided by research in learning theory, should incorporate active student participation, and utilize multiple instructional strategies to maximize student success in mathematics. The use of technology in a mathematics classroom should help students become active learners and develop a deep understanding and profound appreciation of mathematics. Instructional strategies and delivery formats should be designed to respond to different learning and teaching styles, to use technology appropriately to enhance learning, to provide opportunities of access and academic success for all students, and to facilitate avenues for collaboration and cooperation.
Implementing the Standards 

InstructionMathematics faculty will use a variety of teaching strategies that reflect the results of research to enhance student learning. 

At a standardsbased institution, the faculty: 









At a standardsbased institution, the mathematics department and the institution: 



