Thursday 7 March 2024

On "Reducing" Fractions


Just read a note from a young educator who was indignant that a visiting math teacher in her class had used the term "reduce fractions" in a dialog with her student. She wondered if she should "correct" this much senior teacher for her misuse of appropriate edu-speak.

I am no judge of classroom etiquette, but I have written about the history of the usage of the related terms,and thought it might be time to share with the many who may not be aware of the history of the mathematical language of fractional operations.
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 (6) by Lennes, Rogers and Traver, (1942) the term reduction appears in the index only as a subheading under "fractions". The first occurance 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 ducere 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.

And if you like the obvious reductions shown at the top, here is another that is credited to Ed Barbeau who presented “this little beauty of a howler” in the January 2002 College Mathematics Journal, citing Ross Honsberger of the University of Waterloo in Ontario. (With a HT to Greg Ross) Proof left to the reader...

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