One often hears talk about developing conceptual understanding or procedural skill as if they were a dichotomy.  Textbook analyses of textbooks and curricula in selected Asian countries devote a great deal of space to developing the concepts underlying the procedures before or at the same time procedures are developed, insuring that procedures are learned with conceptual understanding (e.g., Mayer et al; Jinfa Cai; Tad Watanabi; Yeping Li; Submitter of this proposal). The Common Core State Standards (CCSS) have referenced Asian curricula (NRC, 2008), suggesting we have much to learn from Singapore Mathematics, e.g.  Forty-eight states have now adopted the CCSS.  The learning of fractions is problematic for many elementary and middle school students and their teachers in the United States.  Researchers in this symposium will report the results of the content analysis in the domain of fractions in elementary mathematics textbooks from Korea, China, Japan and Singapore.   Practitioners using Asian curricula in the United States will report on learning strategies they are using for teaching fractions.  Researchers on the panel will relate their findings to the Common Core State Standards. 

Methodology.  

Content analysis in the domain of fractions has occurred by examining the topics, procedures, and sequencing of the content and procedures in the student texts of one series each from Korea, China, Japan and Singapore.  The teacher manuals from Korea have also been examined extensively.  Major questions in the symposium: 

1.  How is the conceptual and procedural development of fractions treated  in a) Korean text
          books, b) Chinese textbooks, c) Japanese textbooks, d) Singapore textbooks
2. Which concepts and in what sequence are the concepts treated in each text?
3. How is coherence built into the procedural teaching of fractions? a) sequencing in a block
          fashion or in a spiral fashion?  b) multiple representation of number within grade levels?
          c) Consistent representation of number across grade levels?
4. How have the Common Core State Standards (CCSS) incorporated Asian development
          of fractions?  Do the CCSS provide educators tools for developing the conceptual  
          content of fractions from Asian curricula?  Or have the CCSS limited their reference to
          Asian curricula to sequencing of fractions topics in specific grade levels? 

Preliminary Results.

            Korean texts present concepts in a block structure, extending the concepts sequentially in each lesson with little review.  The concept of a fraction is introduced in the second grade by equally dividing a continuous measure. In third grade, students extend the concept of a fraction by an equal division of a discrete measure and understand the relationship between a unit fraction and a proper fraction. In fourth grade, students increase their understanding of  the fraction as a ratio through the concept of multiplication of the fraction, Students inductively learn equivalent fractions through equal division of a continuous measure and multiple equivalent number lines; they observe the relationship of these and find the characteristic of a fraction.  In fifth grade, students study fractions as quotients. Using the fact that multiplying numerator and denominator by the same non-zero number does not change a fraction's value, they find common denominators and reduce fractions, and understand irreducible fractions. They compare fractions with different denominators. Using the relationship between a decimal and a fraction, they convert one to the other. They base the four operations of a fraction on this relationship, and understand and synthesize fractions and decimals as rational numbers.

Chinese 3rd grade textbooks introduce fractions (many unit fractions like ½, ¼). However, the term of "unit fractions" is not been mentioned. In volume 1 a continuous object is used to represent the "whole".  In volume 2 a set of discreet objects is used to represent the "whole".  At this stage, the concept of fractions is conceptualized as "part-whole." This lays a foundation for the in-depth  teaching of fractions at 5th grade. In 4^th grade decimals are taught. 
Volume 1 in Chinese 5th grade textbooks formally introduce  the definitions of “whole,” "fractions" and "unit fractions" (or fraction unit). Similar to the Korean textbooks, "unit fractions" are emphasized. For example, the questions are frequently posed: how many fraction units are contained in a fraction (_____ 1/4s are in 3/4 or 3/4 contains ____1/4s). At this stage,   fractions are conceptualized from an "iteration" approach where multiplicative thinking is involved. Following this approach to conceptualization of fractions, improper fractions are discussed.

Volume 1 in 5^th grade also introduces the basic property of fractions, equivalent fractions.
The activity of "folding" papers is a "splitting" approach that  emphasizes multiplicative thinking.
Volume 1 in 5^th grade also introduces addition and subtraction of fractions with unlike denominators.  The Chinese 6^th  grade textbook presents more advanced topics such as fraction multiplication, division, and mixed operations of fractions.      

The conceptualization of fractions taken by the speaker on Chinese textbooks will focus on the unit fraction and the iteration approach (multiplicative thinking).which is similar to the Korean textbook presentation, aligns well with the recommendations by current fraction literature, and provides the foundation for advanced  topics such as improper fractions, equivalent fractions, and fraction operations.  The speakers on the Japanese and Singapore curricula will focus on strategies they are using or observing in elementary and middle school classrooms in the United States.  Other discussion will focus on the relationship of the above findings to the Common Core State Standards. 

            The symposium panel will consist of six speakers, including researchers, practitioners and the discussant who is widely published on the topic.  The time line has been portioned below.  However, it is possible that we will have the panelists interacting during the entire presentation time, and the interaction extended to the audience well in advance of the last quarter hour of the session. 

Timeline.

The 90 minute timetable will be divided thus:

Korea                                       20  minutes

China                                        20  minutes

Japan and Singapore                20  minutes

Discussant                                 15 minutes

Reaction from audience:            15 minutes