NAMING AND THE TEACHING OF MATHEMATICS

The problem of communicating a thought is the problem of selecting a word or string of words so that the target of communication can better understand that thought. For the average person, this quandary is minimized because a perfect understanding on the part of the target is often not required or sometimes not even desired. For a mathematics instructor, however, the problem remains. Part of the difficulty lies in the names of various mathematical concepts used. Some concepts may actually aid intuitive understanding of the underlying meaning of the word, while others may hinder it. In this paper, several standard English words are evaluated on their ability to provide a straightforward, intuitive understanding of an underlying mathematical concept. An argument for addressing the names of mathematical concepts is presented as well as ideas for further research.

Although communication of mathematical ideas is difficult, effective communication is highly important. The salaries of engineers, computer scientists, financial analysts, and others utilizing mathematics in a professional setting demonstrate the economic importance of mathematical skills. The pervasive and growing role of computers in modern society suggests that those skills will also be important in future occupations. Society also stands to benefit from greater mathematical knowledge in other ways as well: Carl Gauss has deemed mathematics “the Queen of the Sciences” and mathematical developments often lead to developments in the other sciences such as the concurrent development of calculus and classical mechanics or the increasingly utilized statistics and game theory (Smith et al. 2001). In order for students to take advantage of mathematical knowledge, they must have proficient instructors who can communicate mathematics effectively.

The names affixed to mathematical concepts influence the manner in which we think about those concepts, especially when they are first introduced. The term “polysemy” means the coexistence of many possible meanings for a word or phrase. When the name of the concept is first encountered, the student will attempt to conceive the expression in terms of his or her already developed understanding of the standard English meaning of the word. When this previously developed understanding is at least partially aligned with the meaning of the mathematical concept, some intuition for the word’s meaning has been developed by the name alone. The more closely aligned the previously developed understanding and the meaning of the mathematical expression, the more intuitive and apt the name.

Mathematics contains a large number of terms exhibiting polysemy, with some terms particularly well named. The naming of various concepts called “neighborhoods” is one example. Sometimes, however, the names of mathematical terms are misleading, as with the word “imaginary” in “imaginary numbers.” Yet other names range from good efforts (“normal”) to relatively nonsensical (“power”). In this paper, each of the terms mentioned above are investigated in detail, with its ability to shed light on the subject evaluated. The purpose of this research, besides acting as a small survey, is to make instructors and students aware of the role of naming in mathematical understanding and to ideally improve mathematics instruction.

1. NEIGHBORHOOD

The naming of the mathematical concept “neighborhood” is an example of a standard English word providing a good intuition for understanding the underlying mathematical concept. The concept of a neighborhood as commonly understood first appeared in print in 1891 (Neighborhood Def. 5a.). The most common mathematical understanding of neighborhood is: All points lying within a nonzero distance of a given point are in that given point’s neighborhood. This can be compared to the intuitive notion that all the houses on your block are in the neighborhood of your house. The earliest appearances of “neighborhood” in the English language—around the year 1425—have a definition that fits the contemporary understanding of the word: “The people living near to a certain place or within a certain range” (Neighborhood Def. 1a.). The similarities between the Standard English and mathematical usage of “neighborhood,” however, do not lend complete insight into understanding the mathematical concept.

Our intuitive understanding of a mathematical neighborhood breaks down when the definition of neighborhood is presented rigorously. Rudin gives the following definition of neighborhood as used in the mathematical field of analysis: “A neighborhood of radius r of a point p is a set consisting of all points Q such that the distance between P and Q is less than R” (Rudin 28). A neighborhood, thus, has some qualities, size and openness (the distance between P and Q is less than R) that our instinctual conception lacks. Neighborhoods existing in dimensions “higher” than the third dimension are more difficult to understand through the normal English meaning of neighborhood primarily because we cannot visualize such neighborhoods. What is preserved in all dimensions, however, is that “neighborhood” at least vaguely refers to an idea of a spatial relationship, even though such spatial relationships may not be easily visualized. However, this meaning is not the only meaning of neighborhood in mathematics.

Indeed, there are various definitions of neighborhood in different areas of mathematics. In General Topology, John Kelley offers: “A set U in a topological space is a neighborhood of a point X if and only if U contains an open set to which X belongs” (Kelley, 38). Kelley, in contrast to Rudin, omits the idea of a radius and the idea that a neighborhood must be open. In graph theory, we encounter yet another definition, namely that the neighborhood of a vertex is the set of all vertices adjacent to that vertex (Margherita and Weisstein “Neighborhood”). There are other more specialized usages of neighborhood, such as graph theory. Therefore, we must know what type of mathematics we are doing in order to know which type of neighborhood is being discussed, though the type of neighborhood can usually be discerned from contextual clues. These various definitions of “neighborhood” are consistent within each area of mathematics. It is also true that each usage of neighborhood presented preserves the idea of a spatial relationship, thus easing our understanding of each term. If other mathematical terms were named as effectively as neighborhood, students would likely have an easier time understanding mathematics—or at least knowing what a mathematician means when she says she lives in your neighborhood.

2. IMAGINARY NUMBERS

Imaginary numbers, on the other hand, are poorly named. The designation of these numbers as “imaginary” has little to do with the utility of imaginary numbers and more to do with a misunderstanding on the part of those who named them. Although the use of the word “imaginary” denotes some form of uselessness, imaginary numbers have widespread applications in various fields, such as electrical engineering and physics. At the time of their discovery, there were a number of mathematicians, including Euler, Gauss (Nahin 82), and Newton (O.E.D. Imaginary Def. 1c.), who felt that the imaginary numbers were indeed imaginary in the sense of “imaginary or impossible” (Nahin 31). Descartes shared this sentiment and named the imaginary numbers with their unfortunate moniker.

Early reactions against imaginary numbers were not without precedent. Mathematicians have had similar reactions against the concept of zero, negative numbers, rationals, irrationals, transfinite numbers, and many other types of numbers and ideas. As with the early reactions against imaginary numbers, these other responses fell out of favor over time. The ill suited name “imaginary” most likely remains in use due to complacency. Yet referring to such numbers as “imaginary” is not instructive because, in addition to being useful, imaginary numbers have no exceptionally unusual properties. The most surprising properties of imaginary numbers—such as i x i is equal to negative one—are perhaps no more surprising than the properties of real numbers—a negative real multiplied by a negative real yields a positive. Despite this fact, the typical interpretation of the mathematical meaning of “imaginary” as “impossible” persists even in Webster’s Dictionary (Imaginary Def. 1) and there is little doubt it also persists in the minds of students. Since the convention of calling the numbers “imaginary” is so widespread, renaming imaginary numbers is unrealistic. Instructors should explain the inapplicability of the word “imaginary” to their students, which should somewhat mitigate the negative effects of the unfortunate moniker.

3. NORMAL

There are a variety of mathematical terms described as “normal,” many of which are at least somewhat aligned with the everyday English understanding of the word; however, a few terms are not. Barton states that “normal” first appeared in the English language in the sixteenth or seventeenth century “with a mathematical meaning” (Barton 2008) The Oxford English Dictionary (O.E.D.) corroborates Barton’s story, with one exception: three of the four earliest entries for “normal” make reference to the concept of a right angle or a rectangle, but the earliest entry refers to a verb that is “typical” (O.E.D. Normal Def. I. 2a.). The final meaning seems to have been lost; the O.E.D. describes it as rare, although the contemporary English usage of normal as “average” did not appear until 1777 and did not become common until 1840 (Normal Def. 2a.). While the understanding of “normal” as “average” has done much to influence relevant mathematical terms, it has only confused the understanding of the oldest term.

The common meaning of “normal” has influenced the naming of various mathematical concepts. The word has recently taken on specialized mathematical meanings: both in reference to a normal subgroup in algebra and in reference to numbers “having a decimal expansion in which all ten digits, and all sequences of digits of the same length, occur with equal frequency” in number theory, as well as with several more obscure meanings (O.E.D. Normal Def. II. 14b.). The algebraic meaning of normal seems to reflect the importance of normal subgroups in algebra, hence their regular, or normal, occurrence. However, the meaning of “normal” in number theory seems to refer to the regular distribution of numbers described in the term’s definition. The algebraic usage of normal is apparently part of a contemporary trend in the sciences of describing things that occur frequently as normal; such naming has occurred in chemistry, geology, medicine, meteorology, physics, and statistics (O.E.D. Def. II. 6-10).

These names in the context of mathematics do not aid in an intuitive understanding of their underlying concepts. All that they tell the student is that the concepts occur frequently in their respective area of study; they do not help the student understand why they occur frequently or anything regarding the essence of the concept. Instructors should emphasize that the word “normal” has little to do with the meaning of the various mathematical concepts described as normal.

In fact, the oldest mathematical meaning of “normal” comes from geometry. In geometry, “normal” describes angles of ninety degrees or is used in reference to rectangles, shapes containing four angles of ninety degrees. However, these usages of “normal” have fallen by the wayside. In modern geometry, “normal” typically refers to the somewhat related notion of “perpendicular to” (Normal Def. II. 5b.). This idea of “normal” is most commonly observed in three-dimensional spaces in reference to a vector that is constructed tangentially and at a ninety-degree angle—or “normally”—to a plane. As a natural result of being named before the standard English understanding of normal came into common usage, these terms have little to do with the this everyday meaning.

4. POWER

The various mathematical concepts featuring “power” in their names bear little relation to the idea to which the English word “power” generally refers. The first mention of “power” in a mathematical context is found in a 1570 English translation of Euclid’s Elements, which was written circa 300 BCE. Euclid’s meaning of the word “power” means the square of a number, that is, the product of a number and itself. The actual word that Euclid used comes from a word defined as capacity or strength in ancient Greek (Woodhouse Power Def. 1). It is unclear why Euclid chose this word. It also seems that Euclid’s meaning of “power” has been confined to translations of Elements. The common mathematical definition first appeared in 1603, meaning “a quantity obtained by multiplying a given quantity by itself one or more times,” or “an exponent” (O.E.D. Power Def. III. 12c.). This idea generalizes Euclid’s confined notion, to the case where the exponent of a number is equal to two. Although Euclid’s understanding of the word “power” in Ancient Greek appears similar to our contemporary understanding of the word, neither of these definitions seems to have much connection with the contemporary standard English understanding of “power.”

“Power” has a variety of other mathematical meanings: power series found in analysis, power sets in logic and set theory, and the power of a point in geometry. While a power series contains an infinite number of terms raised to a power and the “power of a point” contains the idea of squares of line segments or numbers from Euclid, in a general sense, these designations seem rather arbitrary—almost to the point of being mysterious.

CONCLUSION

Standard English serves the purpose of communication reasonably well, particularly because of its ability to adapt. The names ascribed to mathematical concepts, however, have remained; Euclid’s “power,” for instance, retains a name that is over 2300 years old. Most names ascribed to mathematical concepts, which may have made intuitive sense at the time of their creation, no longer lend insight into the underlying concept—or, as in the case of “imaginary numbers,” never did.

It should be clear that the instructor cannot receive full blame if the student does not understand an idea. As with any subject, mathematics requires effort on the part of the student for full understanding. But given student attitudes toward mathematics, instructors should employ any method that can reduce the difficulty in understanding the subject. Understanding the naming of or potentially renaming mathematical terms exhibiting polysemy is potentially one of those methods.

Consequently, instructors of mathematics attempting to be clear about the names of various mathematical concepts must continually refresh themselves on the significance of those names and devise ways to demonstrate that significance to their students. A catalog of mathematical terms employing an analysis similar to this paper would be useful in this regard. Whatever the method, it should be a priority for instructors of mathematics to make their students aware of the utility of the names of the various words: in at least a small way, the accessibility of mathematics depends upon it.

WORKS CITED

Barile, Margherita and Weisstein, Eric. “WolframMathworld.” Neighborhood. Wolfram Research, 9 Apr. 2010. Web. 26 Apr. 2010.

Barton, Bill. The Language of Mathematics. Boston: Twayne Publishers, 2008.

Devlin, Keith. The Language of Mathematics. Boston: Twayne Publishers, 1998.

“Imaginary.” Def. 1. Webster’s New Twentieth Century Dictionary. 2nd ed. 1978. Print.

“Imaginary.” Def. 1c. The Oxford English Dictionary. 2nd ed. 1989. Online.

Kelley, John. General Topology. Boston: Twayne Publishers, 1975. {:.outdentflush #nahin} Nahin, Paul. An Imaginary Tale. Boston: Twayne Publishers, 1998.

“Neighborhood.” Def. 1a. and 5a. The Oxford English Dictionary. 2nd ed. 1989. Online.

“Normal.” Def. I. 1., I. 2a., I. 5a., I. 5b. II. 6., II. 7., II. 8., II. 9, II. 10., The Oxford English Dictionary. 2nd ed. 1989. Online.

“Power.” Def. III. 12c. The Oxford English Dictionary. 2nd ed. 1989. Online.

“Power.” Def. 1. Woodhouse English-Greek Dictionary. 1st ed. 1910. Online.

Rudin, Walter. Principles of Mathematical Analysis. Boston: Mc- Graw-Hill, 1964.

Sfard, Anna. Thinking as Communicating. Boston: Twayne Publishers, 2008.

Smith, S. A., et al. 2001. Algebra 1: California Edition. Prentice Hall, New Jersey.

BRYCE STUCKI


Bryce Wilson Stucki is a graduating senior from Arlington, Virginia. He is a Mathematics and Economics major with a minor in creative writing and aspires to become a professional baseball analyst. He will be attending a statistics graduate program in the fall to work toward this goal. “Mathematical Polysemy: Naming and the Teaching of Mathematics” is his first published paper. He would like to thank Dr. Daniel Mosser for teaching the class that inspired him to write this paper—and for inspiring his new fascination with linguistics—and his friend Nick Sorenson, who never fails to be a source of intellectual encouragement.