Date: Oct 28, 2012 2:56 PM
Author: Paul A. Tanner III
Subject: Re: The Teaching of Fractions
On Sun, Oct 28, 2012 at 9:21 AM, Robert Hansen <email@example.com> wrote:
> Here is a link to what H.Wu wrote last year regarding the teaching of fractions and the Common Core Standards...
> I disagree with much of it.
How about the Chinese idea of teaching fractions in the 4th grade such that dividing by a number is equivalent to multiplying by its reciprocal, and then using this definition of division in the 6th grade to prove the fraction division algorithm?
In my 1999 edition of "Knowing and Teaching Elementary Mathematics" by Liping Ma we see the following:
Go to pages 58 and 59 and surrounding pages and all the footnotes. But these are not the only pages where these ideas are presented in this book. We find:
Page 58: Chinese elementary school teachers teach to their fourth graders when the concept of fractions is introduced the rather algebraic idea that dividing by a number is equivalent to multiplying by its reciprocal, using this "big word" of algebra "reciprocal", that for instance 9/4 = 9(1/4). This seems to be taught as the definition of a fraction. And note that this is about the reciprocal not of a fraction but of a number. (And what is a reciprocal of a number? It's what you multiply the number by to get 1. And what is 1 called? It's called an identity under multiplication. One would think that they have to teach all this algebraic stuff either then or before then to make it so the term "reciprocal" not of a fraction but of a number actually means what it means.)
Top of page 59: The Chinese 6th grade textbooks (according to this book, 6th grade is still part of elementary school in China - like it used to be in the US until either the late 1970s or in the 1980s) use this idea that dividing by a number is equivalent to multiplying by its reciprocal to justify the division by fractions algorithm. In other words, it mathematical proves the algorithm using what they were taught in fourth grade, which is the definition of fractions as multiplication by the reciprocal of a number. (This is very smart by those who write these textbooks, this giving a rigorous definition of a fraction in 4th grade such that later it is a ready-made means to prove the division by fractions algorithm to 6th graders.)
Top of page 59: In general the Chinese elementary curriculum emphasizes the relationships between operations and their inverses. (With "inverse", there's one of those "big words" of algebra again.)
Chinese elementary school teachers use more "big words" of algebra than "inverse" and "reciprocal" - they use words like "commutative" and "associative" and "distributive". Look at the index and look up all the many references to all these terms, including "fundamental laws" and "subtraction" and "division" and read not only all the text of those pages and surrounding pages, but most especially the footnotes, since much of the meat of these ways of thinking are found presented in the footnotes.
For instance, in the footnotes at the bottom of page 109 of my edition (paperback 1999) we see that third graders are taught the algebraic properties of the commutative and associative laws of addition the abstract way (that's right, using abstract symbols) using letters of the alphabet - we see the following, taught to third graders (that's right, third graders) in the textbooks themselves: "If a and b represent arbitrary addends, we can write the commutative law of addition as:
a + b = b + a."
Notice that not only are they using letters of the alphabet to make a statement that is true for all numbers, they are using the abstract term "addend" here, making the statement even more abstract. In fourth grade they get these algebraic properties stated abstractly for multiplication along with the the distributive law. (See page 132 and surrounding text for a discussion about Chinese textbooks for elementary school.)
At the top of page 111, we see, "From the Chinese teachers' perspective, however, the semantics of mathematical operations should be represented rigorously. It is intolerable to have two different values on each side of an equals side." Read the surrounding text. It compares the US students' way of viewing the equals sign as a "do-something signal" with the Chinese way of thinking of the equals sign, which is more algebraic, as expressed per the above, which is actually the algebraic idea that if we do something to one side of the equals sign, we have to do something to the other side to maintain the equality.
In general, do not miss the footnotes of this book. Read them all including the text and surrounding text they refer to. See that the Chinese way of doing things is a lot more rigorous than the US way, and the term "rigor" applied to arithmetic does have an algebraic component to it since rigor tends to be about generalizing ideas and since algebra is a generalization of arithmetic.