Mathematics communication

Summary: Communication helps mathematicians and others be informed of past and current research and to formulate and organize their own ideas.

Communication is fundamental to mathematics as a discipline, the mathematics community, mathematics education, and society as a whole, since communication is an essential part of everyday life and any social interaction. Effective communication is inherent in validating mathematics. Using a common language and a set of notions and drawing upon a shared body of knowledge, mathematicians communicate with each other—both orally and in writing—about their mathematical ideas, perceptions, or methods.

For example, mathematicians exchange ideas with their colleagues, write technical reports, publish original research papers and expository articles in professional journals, or give oral presentations. Some associate good mathematics communication with beautiful expository lectures or clear writing, while others focus on the quality of the interactions between people, such as those working in a group on mathematics. A peer review process is frequently part of mathematics communication and dissemination, ensuring some degree of consensus on what constitutes appropriate or valid mathematics. In this way, the standards of mathematics are socially developed. In addition to interacting with their colleagues, mathematicians need to communicate with the rest of the society using a language and terminology that are more familiar to the general public. For instance, mathematicians explain to the public how the discipline of mathematics contributes to society or demonstrate the various applications of mathematics in fields such as engineering, medicine, and communication technologies.

The role of communication in the education of mathematics is similar to the vital role communication plays in the discipline of mathematics. Drawing upon mathematical language and notation, teachers and students talk about mathematics; share, explain, and justify mathematical ideas; or analyze, discuss, and interpret mathematical concepts. Communication about mathematics and communication using mathematical language do not occur only in the mathematics community or in mathematics classrooms. Regardless of one’s profession, wise decision making in personal lives and participation in civic and democratic life increasingly demand mathematical communication skills. For example, people need to communicate with mortgage companies when buying a house and interpret various mathematical concepts (such as percentage and rate) presented in the media. Thus, communication with mathematics and about mathematics is an essential part of daily life.

Communication Media

In the twenty-first century, there are a wide variety of electronic and print venues for communicating mathematics, and the evolution of electronic media and databases has vastly changed the way people access mathematics. Historically, mathematicians communicated by letters, during visits, or by reading each other’s published articles or books once such means became available. Some mathematical concepts were developed in parallel by mathematicians working in different areas of the world, such as German Karl Friedrich Gauss and American Robert Adrain, who both made advances in the theory of the Normal distribution in the early nineteenth century. Some mathematicians were not aware of each other’s progress because they did not have the venues of communication that are available in the twenty-first century. In an effort to increase the accessibility of mathematics research articles, reviews began appearing in print journals like Zentralblatt für Mathematik, which originated in 1931, and Mathematical Reviews, which originated in 1940. Since the 1980s, electronic versions of these reviews have allowed researchers to search for publications on a specific topic. In 2010, MathSciNet, the electronic version of Mathematical Reviews, listed more than 2 million items and more than 1 million links to original articles. In 2011, the database Zentralblatt MATH listed more than 3 million items from approximately 3500 journals and 1100 serials. Both contain work dating back to the early 1800s. There are also thousands of mathematics journals that are not listed in these collective databases, such as most mathematics education research.

Some mathematicians publish open access drafts of their papers on their personal Web pages before official publication in peer-reviewed and other journals, or in other online settings such as the ArXiv.org e-print archive. Co-authors from around the world can work together using e-mail or other Web-based collaborative tools. Mathematics students, teachers, and researchers often discuss mathematics ideas and share resources on blogs, through online chats, or using other forums. For instance, what began in 1992 as the Geometry Forum was extended in 1996 to become the Math Forum. There are many additional resources for sharing and teaching mathematics content, both in print and in electronic media. Some electronic examples include the National Council of Teachers of Mathematics Illuminations Web site; Wolfram MathWorld, which was developed by Eric Weisstein; and Math Fun Facts, developed by Francis Su. Social and historical context is also often addressed in sites such as The MacTutor History of Mathematics archive, developed by John O’Connor and Edmund Robertson, or Mathematicians of the African Diaspora, created by Scott Williams.

One important question related to online communication is how to represent and display mathematical notation, which is an important part of mathematical validity and understanding. Some Web pages contain fixed images for each equation or graph. Others use Java applets for dynamic display. The Mathematical Markup Language (MathML) is one way to encode mathematics. TeX was created by Donald Knuth in order to typeset scientific and mathematical research. TeX-based software such as LaTeX has become the standard in printing mathematics. Another issue is the validation of online resources, which may be created or published without peer review. On one level, this issue is an extension of the existing issue of peer review for print media, as mathematics journals already employ varying degrees of rigor when reviewing and publishing papers. At the same time, there is in increasing trend of creating printed works from electronic sources or using electronic sources as references, which creates an added difficulty in ensuring the collective accuracy of the body of mathematics communication.

With so many options available, the specific nature of mathematics communication depends in large part on the purpose and intended audience. There are some mathematics publications and communications aimed at a general audience, others aimed at students, and yet others intended for researchers. Mathematicians, educators, and other communications specialists work to match the form and venue of the mathematics communication to the need. Some careers that are regularly involved in communicating mathematics include technical writers or publication editors. The Society for Technical Communication and the Council of Science Editors are two professional associations that address this need. In 2007, Ivars Peterson became the director of Publications and Communications at the Mathematical Association of America, which, like other professional associations, publishes items for both the specialist and the nonspecialist. He previously wrote MathTrek for Science News. In 1991, he received a Joint Policy Board for Mathematics (JPBM) Communications Award for his “exceptional ability and sustained effort in communicating mathematics to a general audience.” He also served as East Tennessee State University’s Basler Chair of Excellence for the Integration of the Arts, Rhetoric, and Science in 2008 and taught a course there called Communicating Mathematics. In a talk on the topic of communication in mathematics, he noted:

The importance of communicating mathematics clearly and effectively is evident in the many ways in which mathematicians must write, whether to produce technical reports, expository articles, book reviews, essays, referee’s reports, grant proposals, research papers, evaluations, or slides for oral presentations.

Communication in Schools

Communication, both oral and written, is an essential part of mathematics education. The act of communication allows students to systematize and incorporate their mathematics thinking and understanding, both for learning mathematical theory and mathematical problem solving. For example, when students communicate their own mathematical thinking and understanding, they are required to rationalize and organize their reasoning and also formulate puzzling or complex questions well enough to present them as clearly as possible to a reader. As a result, the process guides students toward greater insights to their own thinking and learning. Focused reflection, which is conceptually intertwined with communication, helps students to increase the benefits of communicating their ideas with peers, teachers, and others. Written or oral reflections in which ideas are shared among peers, teachers, and others provide students multiple perspectives that sharpen ideas explored. The American Society of Mathematics (ASM), which is also known as the American Society for the Communication of Mathematics, sponsors problem-solving contests and the U.S. National Collegiate Mathematics Championship.

Proofs

One topic that illustrates the importance and the diverse nature of mathematics communication is the notion of proof. Researchers have proposed a wide variety of roles for proof in mathematics, such as establishing the truth of a statement, communicating mathematical knowledge, opening the way for further understandings and discoveries in mathematics, providing new techniques for doing mathematics, and organizing statements into systems of axioms and theorems. Throughout history, proofs and communication via proof have been incorporated in many different ways in mathematics education in the United States. The National Council of Teachers of Mathematics’ (NCTM) 2000 Principles and Standards for School Mathematics emphasized the role of proof in mathematics learning for all students and helped to formalize its curricular importance and place in pre-kindergarten though high school education. Further, as proof became more systematized in K–12 education, some mathematics education researchers began to more deeply explore students’ understanding of the definition or nature of proof, the role of proof as a mode of communication, and peer acceptance of the validity of a given proof, as well as how proof is taught in classrooms.

As the concept of proof came under investigation, an important issue was the conceptualization and the roles of proof in school mathematics. The NCTM defined proof in Principles and Standards for School Mathematics as “arguments consisting of logically rigorous deductions of conclusions from hypotheses.” One element in the definition of proof is the acceptability of an argument as proof, which is referred to as “logically rigorous.” An important question that NCTM’s definition entails is who decides if a proof is logically rigorous enough to be accepted. To conceptualize the definition and identify the roles of proof in school mathematics, mathematics education researchers have referred to the qualifications and function of proof in the discipline of mathematics and investigated how it is implemented in mathematics classrooms.

Research has demonstrated the social nature of argumentation and justification in the classroom and beyond, and communication and validation by peers plays an important role in proof within and outside the classroom. This social dimension of proof is grounded in sociocultural theories of mathematical learning and is believed to reflect the process of becoming a mathematician. Yu Manin argued that within mathematics community, “A proof becomes a proof after the social act of ‘accepting it as a proof.’” Erna Yackel and Paul Cobb concluded that acceptable justifications in mathematics education are interactively constituted by individual teachers and students in each classroom, where the teacher is the representative of the mathematical community. Mathematical justifications and argumentations are regulated by the general expectations and the regulations of the classroom community.

Thus, they are a part of the classroom norms and, more specifically, the sociomathematical norms, which are the extension of general classroom social norms to specifically focus on the normative aspects of mathematical discussions as students participate in mathematical activities. Yackel and Cobb argued that: “Normative understandings of what counts as mathematically different, mathematically sophisticated, mathematically efficient, and mathematically elegant in a classroom are sociomathematical norms.…The understanding that students are expected to explain their solutions and their ways of thinking is a social norm, whereas the understanding of what counts as an acceptable mathematical explanation is a sociomathematical norm.” This idea plays a role in mathematics educator Andreas Stylianides’s conceptualization of proof. He proposed four aspects that are required to consider an argument a proof: foundation, formulation, representation, and social dimension. He presented an example in which an elementary school student constructed a mathematical argument that was founded on definitions of mathematical constructs, formulated using deductive reasoning from these definitions, and then represented verbally. Regarding the social dimension of the proof, although the student’s argumentation was logically rigorous and would have been accepted as a proof in the wider mathematical community, it generated counterarguments among her classroom peers and her argument was not accepted as a proof by the classroom community.

Indeed, the conceptualization of mathematics, in particular the social dimension that is appropriate for school mathematics, requires more research to develop. Mathematical discourse is an important factor in the development of shared understanding of mathematically valid justifications. However, students at various levels, particularly younger elementary school students, may have different levels of understand regarding the rules and norms of mathematical discourse, and understanding is not necessarily shared by all. Thus, as was the case in Stylianides’ study, a valid mathematical argument was not accepted as valid by all students. In such cases, the teacher, acting as an authoritative representative of the mathematical community, could intervene and explain why the argument is indeed valid by broader standards. However, in some ways this action would negate the social dimension aspect that is used to evaluate mathematical acceptability, at least with respect to the classroom environment. Thus, the subtleties in what constitutes a valid argument within a mathematics classrooms and the relation to a teacher’s role as the communicator of other mathematical norms as they acculturate students in the processes of proving need to be explored. It is important to note that teachers need to know when, how, and how much to intervene so as to not play an authoritarian role, thereby creating a learning environment in which students are forced into authoritarian schemes and communication is essentially unidirectional, from teacher to student.

Mathematical Applications in Communication Technologies

In the increasingly digital world of the twenty-first century, the safe communication of information has become a major issue for discussion and research in mathematics and science, in large part because of theft and fraud often perpetrated using new technologies. Mathematics plays an important role in making communication as safe as possible. Cryptology is a technique used to ensure that messages or data are transmitted safely to the receiver. Dating to the substitution ciphers used in ancient Rome and other civilizations, this field has always drawn heavily from mathematics. Research in mathematics and other disciplines, such as computer sciences and engineering, has resulted in an increasingly sophisticated array of coding techniques and technologies, as well as code-breaking methods. Some of the most common and known applications of cryptography include encryption of credit card numbers or passwords for electronic commerce and encryption of e-mail messages for secure communication. Confidentiality, authenticity, and integrity in electronic commerce or communication have become an apparent and sensitive issue for people who engage in online transactions such as buying or selling items online, online banking, and online communications, as well as for applications like medical records. If proper action is not taken for data transmission, information sent over an open network can be stolen by hackers. Such an action can reveal secret information or messages containing personal information, like a credit card number, a password, or online banking information, facilitating crimes like identity theft. A hacker can use digital data to clone a person’s identity and use a victim’s resources for the hacker’s own good. Even worse, this information could be a national secret, and it may cause more serious problems. For that reason, the National Security Agency (NSA) uses its cryptologic heritage in the midst of challenging times to protect national security systems, and the NSA is one of the leading employers of mathematicians in the United States at the start of the twenty-first century.

Along with digital security, mathematics also plays a fundamental role in both the hardware and software that make the increasingly wireless, globally connected world possible. The Advances in Mathematics of Communications journal publishes research articles related to mathematics in communication technologies. Mathematicians and mathematical methods contribute to many aspects, including the Internet’s computer server backbone and communications protocols; vast cell phone networks; and smartphones that act as mobile platforms for an array of communications methods, such as voice, text, photo, e-mail, and Internet. Music, movies, dance, art, theater, and many other methods people use to convey ideas to one another involve mathematics as part of the creative endeavor. Humans can communicate with neighbors next door, with people on the opposite side of the world, with satellites orbiting the planet, or even with probes that have been sent into the far reaches of the solar system thanks to mathematics. Some would in fact argue that mathematics is itself a universal language or method of communication.

Bibliography

Elliott, Portia, and Cynthia Garnett. Getting Into the Mathematics Conversation: Valuing Communication in Mathematics Classrooms—Readings From NCTM’s School-Based Journals. Reston, VA: National Council of Teachers of Mathematics, 2008.

Manin, Yu. A Course in Mathematical Logic. New York: Springer-Verlag, 1977.

Mathematical Association of America. “JPBM Communications Award.” http://www.maa.org/Awards/jpbm.html.

National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics, 2000.

Peterson, Ivars. “Writing Mathematics Well.” http://sites.google.com/site/ivarspeterson/workshop1.

Stylianides, Andreas. “The Notion of Proof in the Context of Elementary School Mathematics.” Educational Studies in Mathematics 65 (2007).

Yackel, E., and P. Cobb. “Sociomathematical Norms, Argumentation, and Autonomy in Mathematics.” Journal for Research in Mathematics Education 22 (1996).