The purpose of this assignment is to have you reect on the role of language in your learning of mathematics. Begin by reading the blurbs below. The instructions and writing prompt will follow.
READING 1: From Chapter 1. Mathematics as Language in Kenney, Joan M., et al. “Literacy Strategies for Improving Mathematics Instruction.” ASCD (2005).
According to Schwartz (1996), dividing the elements of mathematics into objects and actions has significant implications for curriculum:
To a large extent the arithmetic curriculum of the elementary school as well as the algebra curriculum of the middle and high school focus on the manipulation of symbols representing mathematical objects, rather than on using mathematical objects in the building and analyzing of arithmetic or algebraic models. Thus, in the primary levels, most of the mathematical time and attention of both teachers and students is devoted to the teaching and learning of the computational algorithms for the addition, subtraction, multiplication, and division of integers and decimal and non-decimal fractions. Later, the teaching and learning of algebra becomes, in large measure, the teaching and learning of the algebraic notational system and its formal, symbolic manipulation. Using the mathematical objects and actions as the basis for modeling one’s surround is a neglected piece of the mathematics education enterprise. (p. 39)
But before we can use mathematical objects to model our surround, we must first acquire them. For many reasons, this is an extremely difficult process. Mathematics truly is a foreign language for most students: it is learned almost entirely at school and is not spoken at home. Mathematics is not a “first” language; that is, it does not originate as a spoken language, except for the naming of small whole numbers. Mathematics has both formal and informal expressions, which we might characterize as “school math” and “street math” (Usiskin, 1996). When we attempt to engage students by using real-world examples, we often find that the colloquial or “street” language does not always map directly or correctly onto the mathematical syntax. For example, suppose a pre-algebra student is asked to symbolically express that there are twice as many dogs as cats in the local animal shelter. The equation 2C=D describes the distribution, but is it true that two cats are equivalent to one dog?
As I observed a 4th grade classroom, the teacher began by discussing whole numbers; she then moved to the distinction between even and odd numbers. When asked to classify numbers as even or odd, one of the students, a recent Hispanic immigrant with limited English skills, consistently marked the numbers 6 and 10 as odd. When asked to explain, he said, “Those whole numbers are not multiples of 2.” Additional conversation between teacher and student did little to clarify the problem until I asked the student what he meant by “whole numbers.” It was only when he answered “6, 8, 9, 10, and maybe 3” that we realized that this student had constructed a mental model of “hole” numbers–that is, numbers formed by sticks and holes–and that this was, to him, a totally consistent explanation. If a number had only one “hole,” like the numbers 6, 9, and 10, it was odd, because the number of “holes” was not a multiple of 2. The number 3 was problematic in this student’s system, because he couldn’t decide whether it was really two holes if you completed the image, or two half-holes that could be combined to make a single one. Though this anecdote may seem bizarre, it richly illustrates the difficulty students have as they struggle to make meaning of the words they hear in the mathematics classroom.
An interesting example of how language can either illuminate or obscure concepts is the difference between the word “twelve” in English and the corresponding word in Chinese, the grammar of which is a perfect reflection of decimal structure. One day, as I was observing the piloting of a manipulative device designed to help students understand place value, I talked at length with a Chinese-born teacher who was using the device in his classroom. He told me that in his native language there are only nine names for the numbers 1 through 9, and three multipliers (10, 100, and 1,000). In order to name a number, you read its decomposition in base 10, so that 12 means “ten and two.” This elegant formalism contrasts sharply with the 29 words needed to express the same numbers in English, where, in addition to the words for the numbers 1 through 9, there are special words for the numbers 11 through 19 and the decades from 20 to 90, none of which can be predicted from the words for the other numerals. To compound the confusion, the English word for 12 incorporates two units of meaning: The first part of the word comes from Latin and Greek expressions for “two,” and the second part is related to an Indo-European root meaning “leave” (Wylde & Partridge, 1963). Thus, an etymological decoding would be “the number that leaves 2 behind when 10, the base in which we do our calculating, is subtracted from it”–far from transparent to the novice learner!
To summarize, in mathematics, vocabulary may be confusing because the words mean different things in mathematics and nonmathematics contexts, because two different words sound the same, or because more than one word is used to describe the same concept. Symbols may be confusing either because they look alike (e.g., the division and square root symbols) or because different representations may be used to describe the same process (e.g., • , *, and x for multiplication). Graphic representations may be confusing because of formatting variations (e.g., bar graphs versus line graphs) or because the graphics are not consistently read in the same direction.
READING 2: From page 1-2 in Barta, James, et al. “Math is a verb: Activities and lessons from cultures around theworld.” Reston, VA: National Council of Teachers of Mathematics, 2014.
Mathematics is best understood as we experience its application within cultures and contexts in which it is applied. Everyone in the world is similar in that we all have a spoken language through which we communicate thoughts and ideas. Within this similarity, however, are unique differences shaped and defined by culture and communicated through the diverse vocabulary, syntax, and semantics of each language. Mathematics too is a language comprised of many dialects–dialects that denote diverse communities using mathematics.
ASSIGNMENT and INSTRUCTIONS
Reflect on your experiences learning math topics. Have there been any words or symbols that were particularly confusing to learn? Can you think of symbols or words that are overly similar or used over and over again to mean different things? What could be done to reduce or eliminate confusion, while still allowing math doers to convey necessary information?
Pick a math word and provide its etymology; that is, analyze the history and meaning of the work as Kennedy did with the word “twelve”, or as we did in class with the word “Echelon”. Do you think the current use of your chosen word still conveys insight into the math object? Should we be using a different word?