Language facilitates thought. The purpose of The Language of Mathematics is to help students learn to read, write, and think mathematical thoughts in the abstract, symbolic, language of mathematics.
Warren W. Esty, Department of Mathematical Sciences
Montana State University, Bozeman, MT 59717
(406) 994-5354 wwesty AT theglobal.net (change AT to @)
Mathematical results are expressed in a foreign language. Like other languages, it has its own grammar, syntax, vocabulary, word order, synonyms, negations, conventions, idioms, abbreviations, sentence structure, and paragraph structure. It has certain language features unparalleled in other languages, such as representation (for example, when "x" is a dummy variable it may represent any real number or any numerical expression). The language also includes a large component of logic. The Language of Mathematics emphasizes all these features of the language (Esty, 1992).
The text explains how modern mathematics says what is says. Obtaining numerical answers is not the focus of the text or homework. Students learn how to justify steps, state (simple) theorems, analyze conjectures, use dummy variables, reorganize statements logically, appreciate the importance of hypotheses, read theorems and symbolic definitions of new terms (which is hard for juniors in Advanced Calculus!), and begin to grasp the concept of proof. Together, these components add up to a thorough discussion of how to read, write, speak, and think mathematics.
Fortunately, mathematical sentences and paragraphs are generally written in a limited number of easily distinguishable patterns. Students who are taught to recognize these patterns find mathematics far more comprehensible than those who are not. Furthermore, their abilities to solve problems and do proofs are much enhanced (Esty and Teppo, 1994).
Most examples come from algebra, functions, and set theory (not trig, geometry, or calculus), but the material is the language itself, which is essential for all areas of mathematics. Since this material is not emphasized in any other course, the course level is hard to peg. Some parts look like a "transition to advanced mathematics" course, but, with this unique approach, many students who regard themselves as "terribly math anxious" do very well with the material (Esty and Teppo, 1994).
For a thorough explanation of how the language is essential to mathematics, see "Language Concepts of Mathematics" (Esty) in FOCUS -- on Learning Problems in Mathematics 14.4 (Fall, 1992) pp. 31-54. For the effectiveness of this course, see "A General-Education Course Emphasizing Mathematical Language and Reasoning' (Esty and Teppo), also in FOCUS -- on Learning Problems in Mathematics, 16.1 (Winter, 1994) pp. 13-35. For an article on grading in the context of this course, see the Mathematics Teacher, 85.8 (Nov. 1992) pp.616-618 "Grade assignment based on progressive improvement" (Esty and Teppo).
Because the organization and emphasis of the material is radically new, the use of the text is not (yet) widespread. Idaho State, Montana State, and Coppin State (Baltimore) have decided that it will be necessary for Elementary Education majors (It was not designed for them, but they seem to have special difficulties with abstract symbolism and this course can cure that). At Montana State it has been successfully offered ten years to general students and three times in the summer to secondary math teachers (who knew the procedures of mathematics, but were not so comfortable with expressing them symbolically. A research paper on this will eventually appear). The course was actually designed with freshman math majors in mind, but, general education students in it found that they could "finally" understand mathematics, so, when the word got around, they became the majority of the audience.
Equivalent courses: Probably no other text yields an equivalent course. The level would be about equivalent to a basic logic course -- but, in "The Language of Mathematics" the logic is illustrated by and selected for mathematics. The course is more sophisticated (abstract) than Algebra II, but the content is not at all like "College Algebra" or "Precalculus." It counts as a "Core" course in mathematics at Montana State and Idaho State. It is somewhat above the level of "Finite Math."
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