
Re: Why?
Posted:
Oct 28, 2012 2:48 PM


On Sun, Oct 28, 2012 at 11:45 AM, Robert Hansen wrote: > > On Oct 28, 2012, at 3:54 AM, Paul Tanner wrote: > >> including all the links and citations including what the Liping Ma book >> says. >> > > What the hell are you talking about?
> This is what Liping Ma wrote... > > "I consider the metaphor that Chinese teachers use to illustrate school > mathematics to be more accurate. They believe that elementary mathematics is > the foundation for their students' future mathematical learning, and will > contribute to their students' future life. Students' later mathematical > learning is like a multistoried building. The foundation may be invisible > from the upper stories, but it is the foundation that supports them and > makes all the stories (branches) cohere. The appearance and development of > new mathematics should not be regarded as a denial of fundamental > mathematics. In contrast, it should lead us to an ever better understanding > of elementary mathematics, of its powerful potentiality, as well as of the > conceptual seeds for the advanced branches." > > Ma, Liping (20090320). Knowing and Teaching Elementary Mathematics: > Teachers' Understanding of Fundamental Mathematics in China and the United > States (Studies in Mathematical Thinking and Learning Series) (Kindle > Locations 30133019). Taylor & Francis. Kindle Edition. > > I wrote essentially the same thing a few years ago, not about Chinese > teachers, about how to best teach mathematics. > > To say I have put a fork in this pretense of teaching algebra to elementary > students would be an understatement. The TIMSS assessments and results are > publicly available. So are the PIRLS assessments. Anyone with the > wherewithal can take the time to study them, as I did. After that, track > down as many textbooks as you can. Then come back and tell me that these > countries teach algebra in elementary school. >
Do you deny that that the Liping Ma book says 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 readymade 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 "dosomething 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.

