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.

Characteristics of effective student talks and papers

One of the minicourses of the 2013 Joint Mathematics Meetings asked participants to create a categorized list of characteristics of effective communication in various contexts. To see the results, click here:

Draft characteristics of effective communication from JMM 2013 minicourse

For the first 4 contexts, the topic of communication is mathematics; for the 5th, it is anything quantitative.

These lists have been combined with others created in the MIT Department of Mathematics to create one general list of characteristics of effective mathematics papers and talks. The result (below) reflects the communication priorities of over 30 mathematicians at a variety of institutions.

Combined list of characteristics of effective student talks and papers

The math is correct.

The paper is mathematically correct. For example,

  • In a proof, the hypothesis and assumptions are clearly identified and are differentiated from the conclusion. The logic through which the latter is obtained from the former is explained and is correct.
  • In a mathematical model, appropriate assumptions are made and their consequences are correctly identified.
  • Reasoning (logical/mathematical/quantitative) is correct.
  • Established notation and terminology are used correctly.
  • Theorems are applied only when the premises are met.

The constraints of the assignment are addressed.

  • The paper/talk answers the assigned question.
  • The paper meets the page limit or the talk ends on time. If there isn’t enough space/time to present all of the material, thoughtful choices are made about which material to omit.
  • The talk has been practiced or the paper has been proofread and spelling has been checked.
  • The paper or talk is appropriate for the assigned genre. For example,
    • A research talk/paper begins by placing the research within the context of the field and concludes with suggestions for further research.
    • The degree of objectivity/subjectivity is appropriate to the genre (e.g., research article vs. learning log)
    • The way in which sources and collaborators are acknowledged is appropriate to the genre, as are other conventions such as the purpose(s) of the introduction, the formatting of important statements, and conventions of wording (e.g., “we”) and tone.
    • The balance of originality and use of sources is appropriate.

The audience can perceive the author’s/presenter’s content.

  • Nonstandard terms and notation are introduced before or as they are used. All variables are defined.
  • Figures are self-contained (e.g., axes are labeled, including units of measure if appropriate, and a legend is included if needed).
  • Nothing prevents the audience from focusing on the content (e.g., major issues of grammar, formatting, or writing/presentation style)
  • Type/handwriting/visuals are legible and sufficiently large, including subscripts, labels, etc.
  • For presentations The speaker does not block what is written on the board, either by standing in the way or by blocking boards with other boards. Slides remain visible for long enough.
  • For presentations Speech is sufficiently loud at all times—for example, the speaker does not become too quiet at the ends of sentences, and the speaker faces the audience when saying important points. If the speaker’s speech may be difficult for the audience to understand (due to stutter, accent, etc), strategies are used to ensure that the audience has other means for obtaining important content (e.g., more writing on the board, a handout, etc.)
  • The author/presenter makes clear the extent to which ideas and presentation of ideas are due to external sources.

If the target audience pays attention, they will understand the content.

The author/presenter is aware of the knowledge level of the target audience and anticipates what the audience will need in order to understand the material. For example,

  • The focus of the paper/talk is at an appropriate level of difficulty for the audience, as are the level of formality and the use of technical language. The level of the talk/paper is consistent or, if the audience is varied, the level may progress intentionally.
  • The author/presenter helps the audience to understand the central problem/question and to understand difficult concepts used in the solution; e.g., by anticipating and addressing possible audience confusion, by providing well-chosen examples and conceptual explanations to reinforce concepts, by using a picture iff it is worth a thousand words, and by carefully ordering information to build understanding, and by using words to explain &/or motivate that which is demonstrated with symbols.
  • The author/presenter provides the target audience with all the information they need, including reminders of information that should be familiar but that may not be immediately remembered. Necessary assumptions, definitions, and theorems are carefully stated, and important steps in proofs are stated explicitly. Any gaps in explanations are appropriate for the target audience.
  • At all levels of discourse, clear context and structure help the audience to follow the content. For example,
    • the focus (e.g., the problem, question, or main result)  is clearly stated in the title, in the introduction, and as needed throughout
    • the paper/talk is carefully structured, as is each part of the paper/talk
    • the purpose and relevance of each chunk of detail within the talk is made clear to the audience, for example by outlining or summarizing the approach before going into detail, and by using section introductions, topic and concluding sentences, and other guiding text to indicate context, relevance, and purpose.
    • equations are integrated with text: equations are explained rather than merely listed
    • explanations include “why” in addition to “how”—they are conceptual, not just procedural
    • The reader is guided through the logical development of arguments, which are structured to flow logically. Whenever possible, content is ordered so new content is motivated by &/or follows logically from preceding content (e.g., terms and notation are introduced in context, conjectures are motivated by example).
    • Writing is precise, with deliberate word choice.
    • Formatting of text or structure of writing on blackboards aids understanding (e.g., fractions are built up, equations are aligned, content is grouped logically into boards / slides and information is easy to find).
    • If non-standard notation or terminology is defined by the author/presenter, it is “good” (unambiguous, easy to use, and memorable). The choice of variables makes sense. The use of variables and notation is consistent.
    • For presentations The talk is robust: if an audience member misses an important point or stops to think and then starts listening again later, it is possible to figure out what’s going on (e.g., important points are written on the board, repetition is used strategically, and reminders are given as needed).  Transitions are emphasized and provide sufficient context so those who are lost can start following again.
    • For presentations The presenter monitors audience understanding (for example by asking for questions and waiting for a response), answers questions well, and adjusts the presentation as needed.
    • The introduction &/or conclusion summarizes what has been done/achieved/conveyed, so the audience is not left to determine what conclusions were reached.

In other words, the presentation communicates clearly to the target audience.

Paying attention is easy because the content is presented in an interesting and engaging way.

  • The focus of the paper/talk is chosen based on an understanding of what the audience will find interesting. That focus is used to decide which content to include and which non-essential content to omit; all included content serves an important purpose.
  • The paper/talk begins in a way that makes the audience want to continue listening/reading, and ends in a way that provides satisfying closure.
  • Emphasis is placed on the most important points while subsidiary details are de-emphasized. For example,
    • Examples efficiently introduce basic concepts while formal treatments are reserved for the subtler or more important points that need such treatment;
    • The interesting aspects of a proof are presented while less-interesting details are summarized or omitted;
    • Structure, formatting, pacing, voice, body language, and/or eye contact are used to indicate the relative importance of presented information;
    • An appendix (or post-talk handout) is used for details that are needed for completeness but that are not essential to overall understanding of the topic.
    • The paper/talk is concise and to the point, yet has sufficient explanation and detail for the target audience.
    • The author/presenter elicits interest, excitement, mystery, curiosity, etc., for example by posing well-chosen questions, by commenting about what’s interesting or surprising about the content, and by giving a human dimension to the problem, for example by telling a story from personal experience or from history.
    • Difficult content is presented in such a way that the audience must think, but is rewarded by being able to follow the logic.
    • The full generality of concepts is acknowledged, and creative blending of ideas and forms connects ideas across disciplines.
    • The work is insightful, unique, or elegant.
    • Rather than merely presenting content, the author/presenter guides the audience to discovery.
    • For papers Formatting and guiding text enable readers to actively move around within the paper, referring back to past text and skipping forward to text of interest.
    • For presentations The choice of whether to use slides or blackboards is appropriate for the content and context. Slides/boardwork support delivery without distracting from it (e.g., slide design is simple and consistent with sufficient white space; writing on the board helps rather than hinders pacing of content). The presentation style is enthusiastic and interactive, and demonstrates a solid command of the content. The presenter monitors audience interest (for example by watching facial expressions), and adjusts the presentation as needed.
    • Nothing interferes with audience attention to and interest in the content (e.g., spelling and grammar are correct, language is straightforward, personal bearing is appropriate, use of color is purposeful, there is no gratuitous “showmanship,” and the level of formality and rigor is appropriate for the audience and context, as is the mix of generalities and specifics.)

In other words, the paper/presentation is engaging.

This list can be used to inform both assessment and teaching (e.g., by suggesting questions to guide a class discussion about giving effective presentations).

This list was compiled by S. Ruff based on rubrics by the participants and coordinators of JMM 2013 Minicourse 7 and by H. Miller and on conversations with S. Kleiman, H. Miller, D. Roe, S. Sheffield, A. Nachmias, and J.B. Lewis.

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