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Foundational Information


Math as a Language

Mathematics as a formal discipline of study was developed about 5,000 years ago.

"Now I feel as if I should succeed in doing something in mathematics, although I cannot see why it is so very important... The knowledge doesn't make life any sweeter or happier, does it?" (Helen Keller)

Writing was developed by the Sumerians approximately 5,000 years ago. At the same time, the Sumerians developed some written notation for mathematics. Writing and mathematics are brain tools--they are powerful aids to the human mind. The abilities to use both written language and mathematics are so useful to people that these are "basics" in our formal educational system. Students study and practice the "three Rs" year after year in K-12 education and even on into higher education as they work to develop contemporary and more advanced knowledge and skills (expertise) in these areas.

Our math education system pays some attention to the idea that math is a language. For example, many math teachers have their students do journaling on the math learning experiences and their math use experiences. Some math teachers make use of cooperative learning--an environment that encourages students to communicate mathematical ideas. Some math assessment instruments require that students explain what it is they are doing as they solve the math problems in the assessment.

There has been a great deal of research on the teaching and learning of reading and writing in one's first (natural) language. In addition, there has been a great deal of research on the learning of a second language. It seems likely that some of the research findings and practical implementations of these findings would be applicable to teaching and learning of mathematics.

In the early days of computer programming, there was quite a bit of research done how to identify people who might be good at computer programming. It turned out that music ability and math ability correlated well with computer programming ability. This is interesting from the point of view that in some sense music is a language, and computer programming requires learning programming languages and then solving problems using the languages.


The following article provides some research on the value of directly teaching language skills in various disciplines, including math:

Marzano, Robert J. (September 2005) Preliminary Report on the 2004–05 Evaluation Study of the ASCD Program for Building Academic Vocabulary. Accessed 11/30/05: Building%20

The following email from Garry Taylor is a valuable resource in exploring mathematics as a language.

Subject: your question
Date: January 29, 2005 5:36:13 AM PST

The research seems to say that students can't learn to read mathematics problems better by sensitizing them to vocabulary. We have discovered that when students use manipulatives, and when they describe the attributes and relationships in their own oral language terms, the teacher can, through questioning guide the students to increasingly precise language of mathematics. Once the language is well understood, as are the "definitions" of the concepts and principles, the symbolic translation to equations (numerals and operational symbols) are relatively easy. It all takes time to help students acquire a concept and definitions. Once understood, however, symbolic algorthims seem to take relatively little time to develop--and when asked to solve computational story problems, students seem to have little difficulty in understanding the embedded relationships, numbers and operational procedures to be used.

We have come to this conclusion after 25 years of reading the research on teaching and learning of mathematics. Our direction at the present time is on how this translation occurs.

A few names to understand: Donald Hebb's work on neurological processing (around 1949), his student's work on the same issues; Peter Milner (around the early 1960s; Klausmeier's work on concept acquistion and development (1968 through 1978), Of course Piaget's work has influenced Constance Kamii and others.

I have developed and am in the process of working up a learning model that supports all these positions coupled by more recent work on memory and neurobiology and research into applications of constructivist theory--we are working from the premises of social constructivism.

Garry Taylor, Ph.D.


Music as a language. Quoting Howard Gardner:

“It may well be easier to remember a list if one sings it (or dances to it). However, these uses of the ‘materials’ of an intelligence are essentially trivial. What is not trivial is the capacity to think musically.” (Howard Gardner)

English as a Second or Other Language (ESOL)

Research on Learning Computer Programming and Software Engineering

Mathematics as a Language

Crannell, Annalisa. Writing in Mathematics [Online]. Accessed 1/26/02:

Crannell gives writing assignments in the calculus classes she teaches at a university level. Her Website includes a 1994 booklet A Guide to Writing in Mathematics Classes. Quoting from the first part of that booklet:
For most of your life so far, the only kind of writing you've done in math classes has been on homeworks and tests, and for most of your life you've explained your work to people that know more mathematics than you do (that is, to your teachers). But soon, this will change.

Now that you are taking Calculus, you know far more mathematics than the average American has ever learned - indeed, you know more mathematics than most college graduates remember. With each additional mathematics course you take, you further distance yourself from the average person on the street. You may feel like the mathematics you can do is simple and obvious (doesn't everybody know what a function is?), but you can be sure that other people find it bewilderingly complex. It becomes increasingly important, therefore, that you can explain what you're doing to others that might be interested: your parents, your boss, the media.

Nor are mathematics and writing far-removed from one another. Professional mathematicians spend most of their time writing: communicating with colleagues, applying for grants, publishing papers, writing memos and syllabi. Writing well is extremely important to mathematicians, since poor writers have a hard time getting published, getting attention from the Deans, and obtaining funding. It is ironic but true that most mathematicians spend more time writing than they spend doing math.

But most of all, one of the simplest reasons for writing in a math class is that writing helps you to learn mathematics better. By explaining a difficult concept to other people, you end up explaining it to yourself.

Language and the Learning of Mathematics [Online]. Accessed 1/26/02: A speech delivered at the NCTM Annual Meeting Chicago, April 1988 by Frank B. Allen, Emeritus Professor of Mathematics Elmhurst College. Quoting from the paper:

This brings me to my major thesis that natural language, gradually expanded to include symbolism and logic, is the key to both the learning of mathematics and its effective application to problem situations. And above all, the use of appropriate language is the key to making mathematics intelligible. Indeed, in a very real sense, mathematics is a language. Proficiency in this language can be acquired only by long and carefully supervised experience in using it in situations involving argument and proof.

Mathematics as a Language [Online]. Accessed 1/26/02: Quoting from the Website:

However, the language of Mathematics does not consist of formulas alone. The definitions and terms are verbalized often acquiring a meaning different from the customary one. Many students are inclined to hold this against mathematics. For example, one may wonder whether 0 is a number. As the argument goes, it is not, because when one says, I watched a number of movies, one does not mean 0 as a possibility. 1 is an unlikely candidate either. But do not forget that ambiguities exist in plain English (the number's number is one of them) and in other sciences as well. A a matter of fact, mathematical language is by far more accurate than any other one may think of. Do not forget also that every science and a human activity field has its own lingo and a word usage in many instances much different from that one may be more comfortable with.

The Language of Mathematics [Online]. Accessed 1/26/02:

This Website is based on a book by Warren Esty and a course at Montana State University by the same name. The first quote given below is from the Website, and the second is from the Warren Esty book.
Jointly with Anne Teppo, Warren Esty published an article in the Mathematics Teacher (Nov. 1992, 616-618) entitled "Grade assignment based on progressive improvement" which was reprinted in the NCTM's Emphasis on Assessment. and posted on the web by the Eisenhower National Clearinghouse for Mathematics and Science Education. In a language course, you can expect continual improvement. This article discusses why grading should not be based on averages of unit-exam scores and how a course like "The Language of Mathematics" can be graded.

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 Language of Mathematics [Online]. Accessed 1/26/02:

This Website contains a number of quotations that relate to the topic of mathematics as a language. Here are two examples:
Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say. Bertrand Russell, (1872-1970) The Scientific Outlook, 1931.

The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. Eugene Paul Wigner, (1902-1995): The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

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