The Common Denominator: The Language of Mathematics

Often times, people associate mathematics with numbers and symbols - not words. This association is partially responsible for the difficulties many primary and secondary students have when trying to solve word problems or deal with layers of abstractions when studying algebra. This sentiment is echoed in a paper entitled “Learning the Language of Mathematics,” by Robert Jamison of Clemson University. In it he writes, “Systematic thought does not mean reducing everything to symbols and equations even when that is possible. Systematic thought also requires precise verbal expression. Since serious mathematics is usually communicated in the definition-theorem-proof format, the first step in learning the formal communication of mathematics is in learning definitions.

In my estimation, 80% of the students I taught in 2005 had difficulty negotiating math problems for two reasons: One, they had trouble adding, subtracting, multiplying and dividing; Two, they had trouble understanding the language of mathematics. While it was easier to spot the challenges associated with the fundamentals, it was less obvious discovering student’s difficulties with math comprehension. To make matters worse, parents or guardians would make statements similar to the following: “Math is hard,” or “I’m not surprised you don’t do well in math, I didn’t like math either when I was in school.” All of which creates barriers to a student's success.

I like what was written by U.S. Department of Education, in a pamphlet called “Helping Your Child Learn Mathematics With activities for children in preschool through grade 5.” It says, “ To communicate mathematically means to use mathematical language, numbers, charts or symbols to explain things and to explain the reasoning for solving a problem in a certain way, rather than just giving the answer. “ This seems to suggest that not only should we use numbers and symbols, but also mathematical language.

Based on this premise, we should take a page out of linguistics, the scientific study of human language, to develop a better understanding of how we should approach learning the language of math. In the study of linguistics, important topical divisions are between the study of language structure (grammar) and the study of meaning (semantics and pragmatics).

Grammar encompasses morphology (the formation and composition of words), syntax (the rules that determine how words combine into phrases and sentences) and phonology (the study of sound systems and abstract sound units). Phonetics is a related branch of linguistics concerned with the actual properties of speech sounds (phones), non-speech sounds, and how they are produced and perceived.

However, the basic, most fundamental component of any language is its vocabulary. Therefore, in order to help students master (not learn) mathematics we should, at the very least, focus a portion of the curriculum around the vocabulary.  Moreover, at the introduction of every new math concept, it should be accompanied by a corresponding set of words and their definitions. 

In many instances, especially in the beginning of algebra, you will find the following language:  "Letters such as x and y in a mathematical expression are called variables. Variables are symbols that are used to represent unspecified numbers. Any letter may be used as a variable. An algebraic expression consists of one or more numbers and variables along with one or more arithmetic operations. In multiplication expressions, the quantities being multiplied are called factors, and the result is the product. An expression such as xy is called a power. The variable x is the base and y is called the exponent. The exponent indicates the number of time the base is used as a factor"  I would also take a step back to ensure, more fundamentally than that, the student understand the basics.

A good test you can use to determine whether or not your child or student is having difficult with the language of math, just simply ask the following:  Can you describe the difference between the mode, mean, and median?  Another quick test would be to have them tell you the difference between an integer, whole number, rational, or irrational number.  In many cases you will find that they either confuse the terms or do not know them at all.  However, if you can get to the bottom of it early, you may be able to start seeing some improvements immediately.  At  any rate, let me know the results. Leave a comment or ask questions.