Saturday, 8 October 2022

#11 fraction From old math terms notes

 Fraction comes from the Latin word frangere, to break. A fraction, then, originally represented the broken portion of some whole. The first known use of the word in English is by Geoffrey Chaucer in 1391 in the work, A Treatise on the Astrolabe. Although mostly remembered today for The Canterbury Tales, he was far more famous in his lifetime for this scientific work.  The treatise is considered the oldest work in English written on a complex scientific Instrument. 

By the middle of the 19th Century fraction was used to describe parts larger than the whole as well. In the 1876 edition of Davies' Practical Arithmetic he lists as Article 114. "There are six kinds of fractions:" He then goes on to define

"1. A Proper Fraction is one whose numerator is less than the denominator" Proper fractions are often called "vulgar factions", or common fractions as the term vulgar in Latin referred to "characteristic of or belonging to the masses."  
2. An Improper Fraction is one whose numerator is equal to, or exceeds the denominator."
"3. A Simple Fraction is one whose numerator and denominator are both whole numbers." (Note this is not necessarily what modern teachers would call in "simplest form", for example 8/4 is a simple fraction)

"4. A Compound Fraction is a fraction of a fraction or several fractions connected by the word of or x. The following are compound fractions: 1/2 of 1/4, 1/3 of 1/3 of 1/3, 1/7 x 1/3 x 4."

"5. A Mixed Number is a number expressed by an integer and a fraction." 

"6. A Complex Fraction is one whose numerator or denominator is fractional; or, in which both are fractional," In the Fourth Yearbook of the NCTM in 1929 one of the curriculum changes listed for the State of New York included in the list for the 1910 syllabus, "Fractions, including complex fractions of the 'apartment house' type." (page 161) I assume the "mixed number over a mixed number" is the type of problem referred to, but am still trying to find confirmation of this.

Many modern elementary teachers get upset by the use of the term "reduce a fraction". I think this is mostly because they are not familiar with the origin of the term and only understand the word "reduce" to mean "make smaller", which is certainly one of the most common definitions of the word in modern dictionaries. I hope the the following will make them more understanding of those of us who are VERY old, and still remember when the term had a broader meaning.

According to the OED, the first use of the term in the sense of reducing a fraction was in 1579 in a book by Thomas Digges. Reduction is defined in the 1850 edition of Frederick Emerson's North American Arithmetic, Part Third, for Advanced Scholars as "the operation of changing any quantity from its number in one denomination to its number in another denomination."(pg 29) On the following page it asks the student to "reduce 7 bushels and 6 quarts to pints.". Later in the section on fractions it defines, "Reduction of fractions consists in changing them from one form to another, without altering their value." This broader language is preserved in most later texts for the next seventy or so years. It is defined in Milne's Progressive Arithmetic (1906, William J Milne) thusly, "The process of changing the form of any number without changing its value is called reduction." An almost identical definition appears in Davies and Peck's 1877 Complete Arithmetic, Theoretical and Practical(page 84, art. 66). All the books include reduction of fractions to higher terms as well as lower terms, and reduction of "decimals to common fractions".

In the Late 1930's and 40's arithmetic textbooks seemed to have totally omitted the broader definition, and treat reduce as a vade mecam for fractions in "lowest terms" or "simplest terms". In Learning Arithmetic  by Lennes, Rogers and Traver, (1942) the term reduction appears in the index only as a subheading under "fractions". The first occurrence in the text, on page 36, without prior definition introduces students to a set of problems with the directions, "Reduce the fractions below to simplest forms". In Making Sure of Arithmetic by Silver Burdett (1955) the word "reduce" does not appear in the index at all, but on page 8 it contains, "When the two terms of a fraction are divided by the same number until there is no number by which both terms can be divided evenly, the fraction is reduced to lowest terms." [emphasis is from text]. By 1964, The Universal Encyclopedia of Mathematics by Simon and Schuster contains "A fraction is reduced, or cancelled, by dividing numerator and denominator by the same number." (pg 364) Later on the same page they note, "a fraction cannot be reduced if numerator and denominator are mutually prime" indicating that when they said "the same number" in the first statement, they meant a positive integer. This definition leads to "reduction" of fractions as making the numerator and denominator both smaller.

The roots of the word reduce are from the Latin re for back or again, and dicere which means "to lead". The latter root is also found in the word educare which is literally, to lead out, and is the source of our modern English word, educate.

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