Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Presenting to learn: learning math by talking about it

Most of this site is about “learning to communicate” math; this page is about “communicating to learn” math. In other words, students can improve their understanding of math by communicating about it. The following resources describe or illustrate how giving presentations or talking about math can help students to learn math. There’s another page about writing to learn math.

Many of these resources advocate group work. Information about teaching teamwork (including forming teams, teaching teamwork strategies, and grading teamwork) is available on the page about teaching informal communication. Many of the following resources were found by undergraduate researcher Noor Doukmak.


In many educational models, students teach themselves some (or all) of the math rather than solely listening to lectures. Such learner-based models often involve communication in the form of  collaboration, in which students discuss math with each other, and/or presentation, in which students demonstrate what they’ve learned &/or teach it to other students. The specific names applied to these learner-based strategies vary over time (and space). See below for some examples.

  • “Inquiry-based learning refers to any pedagogy that replaces traditional lectures and textbooks with some form of student-centered activities…The Journal of Inquiry-Based Learning in Mathematics (JIBLM) publishes university-level course notes that are freely downloadable, professionally refereed, and classroom-tested.”
  • The Academy of Inquiry-Based Learning provides videos, workshops, mentoring, small grants, etc.
  • The Educational Advancement Foundation“supports grants of various amounts in line with its mission”:
    • “the development and implementation of inquiry-based learning at all educational levels in the United States, particularly in the fields of mathematics and science, and
    • “the preservation and dissemination of the inquiry-based learning methodology inspired by Dr. R. L. Moore…”
  • The website The Legacy of R.L. Moore provides demonstrations, a quick-start guide, and literature about the method R.L. Moore used with graduate students. In the Moore method, selected students worked individually to prove theorems and presented their work to each other.
  • To enable the Moore method to be successfully used in undergraduate classes, modifications have been made that include having students work together in small groups rather than individually, and meetings with the instructor in which the instructor can guide inquiry. See these articles about the Modified Moore Method.
  • The Moore method and inquiry-based learning often center around tasks that are “small” enough for several solutions or proofs to be presented in one class session. In contrast, in some undergraduate seminars, students give the lectures and only one or two students lecture in each class session. See the page about Undergraduate Seminars; some articles are listed at the end of the page.

Myriad other courses replace or supplement lecture with active modes of learning that involve student communication. Some resources and specific examples follow:

  • A Practical Guide to Cooperative Learning in Collegiate Mathematics, by N.L. Habelgans et al., MAA Notes #37, 1995.
  • Readings in Cooperative Learning for Undergraduate Mathematics, edited by E. Dubinsky, D. Mathews, and B. E. Reynolds, MAA Notes #44.
  • GoodQuestions Project, Department of Mathematics, Cornell University
    This website includes concept questions for checking understanding and generating discussion in calculus classes. From the website: “The GoodQuestions project seeks to improve calculus instruction by adapting two methods developed in physics instruction — ConcepTests and Just-in-Time-Teaching. GoodQuestions is a pedagogical strategy that aims to raise the visibility of the key concepts and to promote a more active learning environment.”
  • “From Calculus to Topology: Teaching Lecture-Free Seminar Courses at All Levels of the Undergraduate Mathematics Curriculum,” by D. L. King, PRIMUS, Vol 11, No 3, Sep 2001, pp. 209-227.
    From the abstract: “The paper discusses the author’s experiences teaching mathematics courses in a lecture-free, discussion-based seminar format at all levels of the traditional undergraduate mathematics curriculum. These seminars replace the traditional lecture with classroom discussion of course readings and exercises prepared by students in advance of class.”
  • “Mathematics Through Workshops Based on Collaborative Learning,” by A. B. Kasturiarachi, PRIMUS, Vol 7, No 2, Jan 1997, pp. 147-163.
    Abstract: “…The goal in this article is to demonstrate, based on the Academic Mastery Program (AMP) at Occidental College, how to incorporate collaborative learning into the mathematics curriculum. AMP is a program…[for]…freshman undergraduates…The structure of the program, we believe, minimizes the isolation of women, minorities, and borderline students by transforming each student’s contribution into the success of a group. We will describe in detail the format of the weekly two hour workshops, which are supervised by student facilitators. Issues such as student recruitment, hiring facilitators, and designing worksheets will also be discussed.”
  • A Handbook for Mathematics Teaching Assistants” by Tom Rishel, The MAA
    This handbook includes brief but helpful guidance on cooperative learning, writing assignments, and the active classroom, as well as links to further resources.
  • Models in Algebra and Rhetoric: A New Approach to Integrating Writing and Mathematics in a WAC Learning Community” by Ronald J. Heckelman and Will-Matthis Dunn III, Language and Learning Across the Disciplines 6(3) August 2003.
    This paper describes the integration of college algebra with rhetoric & writing in a team-teaching situation. “We make no claims that learning to do algebra or to write becomes “easier” as a result of the focus on models…Nevertheless, our students have demonstrated that yoking the two disciplines by focusing on models provides a powerful critical instrument…”

Secondary School

  • The IMPlementation Center: Interactive Mathematics Program Resource Center“, The Interactive Mathematics Program (IMP), Key Curriculum Press
    This website presents a “four-year program of problem-based mathematics that replaces the traditional Algebra I-Geometry-Algebra II/Trigonometry-Precalculus sequence…”
  • Complex Instruction: Raising Achievement Through Group Worthy Tasks,” NRICH enriching mathematics
    This website provides strategies for including groupwork in math classes successfully. Provides strategies for encouraging all members of the team to contribute, including choosing group-worthy tasks, assigning group roles, rewarding diverse contributions to groupwork, and enabling quieter students to contribute by raising their status within the group. Videos demonstrate these techniques. A workshop summary and discussion about Complex Instruction is on the blog Math Mama Writes… by Sue VanHattum.
  • Engel, J. S., “Students Questioning Students (SQS): a Technique to Invite Students’ Involvement,” Gifted Education International, September 1988, Vol. 5, No. 3, pp. 179-185.
    Secondary students teach the classes not via lecturing but via questioning their classmates. A video about SQS is also available.
  • Huang, J. and B. Normandia, Students’ Perceptions on Communicating Mathematically: A Case Study of a Secondary Mathematics Classroom, The International Journal of Learning, 2009, Vol. 16, No. 5, pp.1-22.
  • Huang, J. et al., “Communicating Mathematically: Comparison of Knowledge Structures in Teacher and Student Discourse in a Secondary Math Classroom,” Communication Education, Jan 2005, Vol. 54, No. 1, pp. 34-51.
    This is a study about students in a pre-calculus class whose teacher had them spend much of class time talking to one another about math content. Each student explains to his peers how he solved a math problem and then listens to his peers do the same.
  • Improving Writing and Speaking Skills Using Mathematical Language (ERIC record)
    From the abstract: “A study investigated the effectiveness of an intervention for improving adolescent writing and speaking using mathematical language… [The intervention consisted of] journal writing, cooperative learning, and portfolio assessment. Results indicated that: (1) students’ attitudes toward writing and mathematics improved; (2) students’ understanding of mathematics improved; and (3) students showed an increase in their ability to talk mathematically.”
    Includes reviews of literature about including communication in math classrooms. Includes appendices with sample writing prompts for prealgebra and algebra.
  • “Integrating Oral Presentation into Mathematics Teaching and Learning: An Exploratory Study with Singapore Secondary Students” by Fan Lianghuo & Yeo Shu Mei, from The Montana Mathematics Enthusiast, Monograph 3, October 2007 Festschrift in Honor of Guenter Toerner’s 60th Birthday, pp. 81-98.
  • Johnson, D. W., & Johnson, R. T. (1990). “Using Cooperative Learning in Math.” In N. Davidson (Eds.) Cooperative Learning in Mathematics: A Handbook for Teachers. (pp. 103-125). Addison-Wesley Publishing.

Elementary School

Classroom Discussions: Using Math Talk to Help Students Learn, Grades K–6, Second Edition by Suzanne H. Chapin, Catherine O’Connor, and Nancy Canavan Anderson
“This best seller offers an unparalleled look at the significant role that classroom discussions can play in teaching mathematics and deepening students’ mathematical understanding.”

Using communication to help reduce math anxiety

Communication assignments are often used as part of a strategy to reduce math anxiety, thus indirectly improving learning of mathematics. See the page of resources and research about using communication to reduce math anxiety.

Professional development opportunities and teacher training

Many teacher workshops address active forms of learning that involve students communicating about math. See, for example, the following opportunities:

Research on the effectiveness of various modes of active learning

Although the various modes of active learning described above provide students with the opportunity to communicate mathematics, their effectiveness for learning mathematics likely depends on a variety of factors. Some relevant research is listed below.

  • Prince, M., “Does Active Learning Work? A Review of the Research,” Journal of Engineering Education, July 2004, pp. 223-231.
  • Laursen, S., et al., “Evaluation of the IBL Mathematics Project: Student and Instructor Outcomes of Inquiry-Based Learning in College Mathematics: A Report Prepared for the Educational Advancement Foundation and the IBL Mathematics Centers,” Assessment and Evaluation Center for Inquiry Based Learning in Mathematics, April 2011.
    This study conducted by the Assessment and Evaluation Center for Inquiry Based Learning included over 100 course sections at four campuses across two academic years. “The approaches implemented at the IBL Mathematics Centers benefited students in multiple, profound, and perhaps lasting ways. Learning gains and attitudinal changes were especially positive for groups that are often under-served by traditional lecture-based approaches, including women and lower-achieving students. First-year and less mathematically experienced students also benefited particularly. Yet there was no evidence of negative consequences of IBL for men, high-achieving students, older and more experienced students: these groups too made gains greater than their non-IBL peers.” More publications from this study are available from the website of Ethnography and Evaluation Research at the University of Colorado, Boulder.
  • Kirschner, P. A., et al., “Why Minimal Guidance During Instruction Does Not Work: An Analysis of the Failure of Constructivist, Discovery, Problem-Based, Experiential, and Inquiry-Based Teaching,” Educational Psychologist 41(2), 75-86, 2006.
    Abstract: “Evidence for the superiority of guided instruction is explained in the context of our knowledge of human cognitive architecture, expert–novice differences, and cognitive load. Although unguided or minimally guided instructional approaches are very popular and intuitively appealing, the point is made that these approaches ignore both the structures that constitute human cognitive architecture and evidence from empirical studies over the past half-century that consistently indicate that minimally guided instruction is less effective and less efficient than instructional approaches that place a strong emphasis on guidance of the student learning process. The advantage of guidance begins to recede only when learners have sufficiently high prior knowledge to provide “internal” guidance. Recent developments in instructional research and instructional designmodels that support guidance during instruction are briefly described.”
  • Bressoud, D. M., “The worst way to teach,” Launchings, July 2011.
    This blog post summarizes some of the research that supports Halmos’ quote, “The best way to learn is to do; the worst way to teach is to talk,” or as Bressoud puts it, “Shut up and teach.”
  • Brickman, P., et al., “Effects of Inquiry-based Learning on Students’ Science Literacy Skills and Confidence,” International Journal for the Scholarship of Teaching and Learning, Vol 3, No. 2, July 2009.
    From the abstract: “In this study, we demonstrated greater improvements in students’ science literacy and research skills using inquiry lab instruction. We also found that inquiry students gained self-confidence in scientific abilities, but traditional students’ gain was greater – likely indicating that the traditional curriculum promoted over-confidence. Inquiry lab students valued more authentic science exposure but acknowledged that experiencing the complexity and frustrations faced by practicing scientists was challenging, and may explain the widespread reported student resistance to inquiry curricula.”

Please suggest good research to add to this bibliography. Of greatest interest are literature reviews, meta-analyses, and studies of multiple courses and institutes, but any relevant research is welcome.

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