You are here

Speaking Mathematically

David Pimm
Publication Date: 
Number of Pages: 
Routledge Revivals: Language, Education and Society Series
[Reviewed by
Valentin A. B. Küchle
, on
Although published over thirty years ago in 1987, David Pimm’s book Speaking Mathematically: Communication in Mathematics Classrooms remains an exceptionally relevant source of insights for mathematics instructors. On the surface, Pimm explores the metaphor “mathematics is a language” for insights about the teaching and learning of mathematics. But what makes the book so powerful is that it pinpoints and discusses moments of confusion for learners of mathematics that instructors may have forgotten with the passing of time. By approaching mathematics from a linguistic lens, the reader cannot help but marvel at the confusing things mathematics instructors say and do. In short, the book provides an opportunity for readers to appreciate the confusing nature of mathematical conventions, increase their awareness of novices’ struggles, and reflect on their own teaching.
With the goal of exploring the metaphor of mathematics as a language and pushing it to its limits, the book focuses on spoken mathematics in chapters 2–4. In these chapters, Pimm progressively widens his scope from students’ talk (chapter 2) to classroom talk (chapter 3) to the words and structures of mathematics and the meanings created by them (chapter 4). In the second half of the book, Pimm shifts the focus to written mathematics: Chapter 5 is dedicated to students’ writing, in chapter 6, light is shone on the mathematical writing system, and, in chapter 7, Pimm explores the idea of mathematical grammar. Each of these chapters is full of stimulating questions and ideas, and I have selected a few that I believe are representative of Pimm’s message and which the reader may find particularly thought-provoking. 
After introducing the book, Pimm begins his discussion of spoken mathematics in chapter 2 by reflecting on the function of talk. Communication and reflection are two examples of functions of talk, which raises the question: What kind of talk helps students learn mathematics? Pimm notes that just talking—or just telling students to talk—rarely results in learning. For talk to support learning, Pimm recommends that it should be focused, explicit, geared towards changing the listener’s state of knowledge, and disembodied (i.e., expressions should be able to stand on their own). Yet, if instructors embrace these expectations, they need to be transparent about their goals and reasons for endorsing them. If they are opaque about their intentions and expectations, instructors may find themselves in confrontation with students who adhere to different norms.
In writing about classroom talk (chapter 3), Pimm introduces the term gambit to mean a move made by a teacher which may involve making a potential “sacrifice” in the hope of gaining an overall advantage. Consider, for example, the gambit of letting students discuss content in pairs: Although this gambit may result in greater class participation since students have practiced their response in a more comfortable setting, instructors relinquish control and awareness over the conversations. Using this sacrifice-advantage lens, Pimm discusses a number of other moves instructors traditionally make and explores alternatives. A “mini-gambit”—my word, not Pimm’s—is instructors’ fondness for the pronoun “we.” Pimm explains how problematic use of “we” is with regards to authority, persuasion, membership, (reduced) responsibility, and fear of exposing oneself. It may be that through the use of the pronoun “we” instructors unintentionally bully students into accepting content by appealing to the authority of the community, play on students’ desire to belong to said community, and reduce their responsibility for clearly presenting content. In short, Pimm makes readers reflect on how (our) use of “we” affects (our) students. More broadly, this chapter asks: What sacrifices do instructors make for the advantages they hope to gain with the gambits they choose? 
In chapter 4, Pimm describes a number of ways in which new words and meanings are created in mathematics. One way is to borrow from everyday language which may result in semantic contamination and lead to an inappropriate transfer of meaning, for example, the expectation that non-open sets are closed or that “or” is exclusive. More generally, mathematics is full of metaphors which seek to highlight similarities across different contexts, yet leave the uninitiated confused (e.g., spherical triangles, clock arithmetic, and the widespread appearance of “+” in many contexts). Pimm warns that the elusiveness of metaphors combined with the absence of discussion of metaphors can be detrimental to students. On one hand, students may fail to exploit the power of metaphors, and, on the other hand, they may unwittingly carry over ideas that are unhelpful or even detrimental to their understanding.
The discussion of students’ writing in chapter 5 bears some resemblance to chapter 2’s focus on students’ talk. Pimm makes it clear that students often do not know for whom and why they are writing. If we are honest with ourselves as mathematics educators, we should ask: Do students benefit when copying word-for-word what is written down in lectures when lecture notes or textbooks are available to them? Pimm concedes that writing can externalize thinking, but questions whether students are able to think while listening to lectures and copying board work. Furthermore, making students copy board work may have adverse effects on students’ self-expectation. If what is written on the board is always perfect, students may feel that they need to be faultless and become paralyzed by those self-expectations. To avoid unfruitful copying of board work and move towards more productive forms of writing, Pimm encourages instructors to be upfront with students—and themselves—for whom and why they are writing.
Two points from the last two chapters on written mathematics seem most central. The first is that the mathematical language is not as universal as we may wish to believe. Consider, for instance, the example of \( F_{\sigma} \) and \( G_{\delta} \) sets. An \( F_{\sigma} \) set is a countable union of closed sets, whereas a \( G_{\delta} \) set is a countable intersection of open sets. Note that F stands for “fermé” (i.e., “closed” in French), \( \sigma \) is supposed to adumbrate “s” which is supposed to evoke “somme” (i.e., “sum”/“union” in French), \( G \) stands for “Gebiet” (i.e., “area”/“neighborhood” in German to evoke openness), and \( \delta \) is supposed to bring to mind “d” which is supposed to evoke “Durchschnitt” (i.e., “average”/“intersection” in German). After hearing this derivation once or twice, someone who speaks French and German and has knowledge of the Greek alphabet can rederive the meaning of \( F_{\sigma} \) and \( G_{\delta} \) sets at will. Now imagine not having access to any of this knowledge. Mathematics is full of examples where the meanings of symbols are motivated for only a select few, typically speakers of European languages. The problem extends beyond symbols to, for instance, the way in which we read and write mathematics (left-to-right, top-to-bottom). In essence, how does the mathematical writing system affect those unfamiliar with European languages? How can they profit from the meaning underlying mathematical writing when it is full of mnemonics and structures which may appear arbitrary to them?
Second, regardless of one’s mother tongue, one may find that mathematics can be rich in symbols and weak in meaning. In particular, Pimm asserts that students often suffer from instructors’ focus on form over content. Mathematical writing is powerful because mathematical objects are represented by easily manipulable symbols. Algebraic manipulations are particularly potent, and it is no wonder that we want students to be fluent and accurate. Yet, the desire for students’ algebraic proficiency rationalizes classes where algebraic manipulations and algorithmic procedures are the sole focus. It is questionable whether such classes allow students to reattribute meaning to symbols—an essential skill for mathematicians. Should student errors like the Freshman’s dream really surprise us? With a nod to Kant, Pimm notes that “while semantics without syntax is blind, syntax without semantics is empty” (p. 178). 
In the final two chapters, Pimm reflects on and synthesizes the content of the preceding chapters. Ideas that surfaced in previous chapters are repeated and woven together. In this vein, I highlight my six takeaways from the book. First, students’ talk should be seen as an opportunity for learning. Yet, in order for students to engage in productive talk, instructors need to be transparent about their goals and motivations when establishing the norms of student talk in the classroom. Second, any teaching move comes with advantages and sacrifices. Whatever move you choose (e.g., use of the pronoun “we”), be cognizant of its advantages and your sacrifices. Third, mathematicians have often borrowed words from everyday language. Although a fluent speaker of mathematics may imbue these words with mathematical meaning, we should remember that for students the term initially possesses only its everyday meaning. Fourth, although writing can serve to externalize thinking, not all writing supports thinking. For students to profit from writing, instructors should be honest and transparent about why and for whom students are being asked to write. Fifth, European languages have had a major influence on the symbols, structure, and syntax of mathematics. What implications does this have for speakers of other languages? What could be done to level the playing field? Sixth, there is power in encapsulating objects in symbols, manipulating the symbols, and de-encapsulating symbols to arrive at conclusions about objects. Students, however, are often only taught how to manipulate symbols. Instructors should ensure that form and symbols—and their manipulation—are not given precedence over meaning. 
If none of the above points have convinced you to pick up Speaking Mathematically: Communication in Mathematics Classrooms, let me end by saying that I have not been able to do the book justice. Reading this book will sensitize you to the language of mathematics: to what you say, to what you hear, to what you write, to what you read. Reading this book will change you, and it will very likely change how you engage your students.
Acknowledgement: I am indebted to Dr. Jack Smith for his support in crafting this book review. 
Valentin A. B. Küchle is a mathematics education graduate student at Michigan State University who does research in undergraduate mathematics education. His interests include mathematics instructors' beliefs about teaching and learning, studying proof as a genre of writing, the development of students' agency and autonomy in proof-based classes, and students' understanding of ring theory. At the moment, he dabbles in commognition and the study of mathematical discourse.

See the publisher's web page.