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“Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples

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Abstract

This article deals with the roles of variation problems (“one problem multiple solution” and “one problem multiple changes”) as used in Chinese textbooks. It is argued that variation problems as an “indigenous” Chinese practice aim to discern and to compare the invariant feature of the relationship among concepts and solutions. This practice also aims to provide opportunities for making connections, since comparison is considered the pre-condition to perceive the structures, dependencies, and relationships that may lead to mathematical abstraction. In the first part of the article, the “indigenous” practice is discussed against its philosophical Daoism and Confucianism backgrounds. To grasp its distinctiveness, a comparison between Chinese and American textbooks is carried out. In the second part of the article, the focus is on the manner in which fraction division is articulated in an important Chinese textbook. A framework to understand variation practice is introduced and some educational implications are suggested.

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Notes

  1. For example, Zodik and Zaslavsky (2008) found that experienced secondary mathematics teachers were largely unconscious of differences in quality of the example choices they made.

  2. Bianshi can be translated loosely as “variation” in English.

  3. Exposed in western colony (Hong Kong & Macau) for years, it is impressive that variation practices are common in Chinese teaching, while they are not important in others when classrooms and curricula are compared.

  4. It is possible, however, that different Asian countries influenced by China share the same components of variation practice or variation problems but lay different emphases and different combinations. China is more typical than others. We believe this discussion will lead readers to reflect more deeply on their own cultural practices, as well as consider variation practice in different cultures in new ways. We believe that considering different points of view would be beneficial.

  5. The “indigenous” definition of Bianshi practices is narrower than that used by Marton and his colleagues.

  6. For example, Marton and Pang (2006) presented a framework to examine variation in detail, consisting of contrast, separation, generalization, and fusion (p. 199), which could be applied to all subjects.

  7. The “indigenous” notion of MPOS (multiple problems one solution, varying representation), a kind of variation practice, will not be used in this study because it is mainly applied to summary exercises.

  8. We also hope this paper will articulate the “typical” practice of problem variations and their roles in Chinese mathematics curriculum, and become a bridge that can bring about connections regarding problem solving or curriculum development, while at the same time providing a window through which more light can be shed on Chinese mathematical practices.

  9. For example, in a survey, 102 Chinese teachers were asked to point out how often they use these variation problem activities in the classroom. Results showed that all the teachers used them, although they used them in different degrees. Over half of the teachers surveyed used OPMS,MPOS,and OPMC quite often in their instruction (Nie, 2004).

  10. The following words of Confucius, “If one can’t respond with the other three corner[s] if a corner is shown, then one is not in the proper track (of learning),” (The Analects, 7:8) (「舉一隅不以三隅反者, 則不復也.」─《論語‧述而第七》) also highlights the idea of discerning the invariant element in variation practice.

  11. Stuctures in other treatises are similliar to those in “Jiuzhang Suanshu” 《九章算術》, such as “Haidao Suanjing” 《海島算經》 “Zhangqiujian Suanjing” 《張丘建算經》 “Wuchao suanjing” 《五曹算經》 “Wujing Suanshu” 《五經算術》 “Figu Suanjing” 《緝古算經》 “Shushu jiyi” 《數術記遺》 “Xiahouyang Suanjing” 《夏侯陽算經》.

  12. E.g., Euclid's “Elements” stressed a deductive system

  13. Zhoubi Suanjing (周髀算经) is a classic Chinese treatise, which is represented as a teacher, Chenzi, telling his student, Rongfang, how to learn mathematics, which scarcely is known to west history. It is worthy of notice that as in the case of the dialogue between Socrates and the slave-boy in Plato's Meno, the first piece of Chinese writing about mathematics is in the form of a conversation between master and student.

  14. It means categories of problems here.

  15. A similar thought, “profound understanding of fundamental mathematics, emphasizing depth, breadth, and thoroughness understanding, as against surface procedural knowledge” was expressed in the study conducted by Ma in 1999 (p. 124).

  16. For example, Ball (1990) reported that few American prospective teachers were able to give a mathematical explanation of underlying principle and meaning.

  17. One of the possible reasons for the ambiguity of the studies above is that “learning is seen as a change in an individual’s way of experiencing the object of learning” (Marton & Booth, 1997). From this perspective, it is difficult to categorize what is not a variation case in learning or teaching, since the pedagogy of every individual is always changing.

  18. The variation practices have spread into a wide range and variety of forms in China; it is not easy to find an indigenous “prototype”, which clearly reflects local “routine” practice.

  19. Students could fail to recapitulate the relationship of addition and subtraction, and the meaning of “equal” from the problem set 1 + 2 = 3, 2 + 1 = 3, 3 − 1 = 2, similar to learning the meaning of addition from a single problem, 1 + 2 = 3. The problem set hinges on exemplifying relationships rather than objects. In this sense, learning is not an action where one remembers something as much as he or she can; it is concerned with discerning the general relationship of addition and subtraction (and that a similar case happens between addition and multiplication, division and multiplication, and so on), then building a comprehensive structure. In this case, the subtraction problem 3 − 1 = 2 should help in understanding addition rather than hampering its comprehension. The disadvantage in “one-thing-at-the-time” design is that the commonality of the part–whole relationship between addition and subtraction is neglected or taken for granted. In fact, an addition problem conveys the same message as a subtraction problem, or it “translates” the same part–whole meaning. In this regard, the addition/subtraction problem reflects two aspects of part–whole meaning, and therefore should not hamper, but support each other.

  20. Marton and Booth (1997) presented a framework to analyze “Variations and invariant”; Marton and Pang (2006) also extended another framework to examine variation in detail, consisting of contrast, separation, generalization, and fusion (p. 199).

  21. For example, Fan and Zhu (2004) reported that almost 90% of the lessons are those in which teachers teach examples, directly embodying depth and duration of each topic in the textbook.

  22. The scope of the article and analysis of the examples is limited to the topic of division by fraction only.

  23. The latest reform version is heavily influenced by Western theories and traditions.

  24. “來源書本, 但高於書本,” words from The Illustration of 2003 National College Entrance Examination for Science Majors in China.

  25. The seven examples analyzed required a single computation, 2-step computation, 3-step computation, and 4-step computation.

  26. All the five examples required a single computation procedure only.

  27. There were six examples to elicit the conceptual rationale in the Chinese textbook but not a single example to elicit the conceptual rationale in the US textbook. There were five examples that demonstrate procedural algorithm in the US textbook and only one example in the Chinese textbook.

  28. For example, “If five items cost $10 and all items are the same price, then I can find the cost of one item by first dividing $10 by 5 to find out how much the item costs. If one of two items cost $10 and all items are the same price, then I can find the cost of one item by dividing $10 by one of two to find out how much the item costs”. The concept of division by fraction is the same as that by whole number.

  29. Four US textbooks tended to use the least number of pages for fraction division (3–15 pages in one textbook) compared with the 26 to 38 pages typically found in Chinese textbooks.

  30. It is common sense for curriculum developers to utilize existing knowledge to elicit new knowledge.But it seems that Chinese generally stress much more than others.

  31. This means there are at least two concepts targeted within a problem.

  32. For instance, example 1 is designed for eliciting multiplication, division, and equation concepts.

  33. These roles indicate the big Chinese ideas relating to curriculum development, which may be traced to the deep-rooted Chinese cultural philosophy, I jing, in which are thoughts on abstracting invariant concepts from a varied situation and applying these invariant concepts to the varied situations (in Chinese變中發現不變, 以不變應萬變).

  34. Schmidt, Houang, and Cogan (2002) suggest that the content feature of a coherent/incoherent curriculum is characterized by highly nonrepetitive/repetitive, focused/unfocused, challenging/unchallenging conceptual organizations.

  35. It should be more demanding for Chinese students to learn fraction division in grade 6 only once than their US counterparts in grade 6, 7, 8, who learn it three times.

  36. In 2006, an experiment was carried out on a treatment group, where a textbook developed with heavy emphasis on relationships with problem variation in fraction division was used in three schools. In the control group, the traditional HK textbook heavily influenced by England principles and having light emphasis on relationships was used in another three schools. The experimental treatment group achieved a better conceptual understanding of fractions, division, and multiplication compared with the control group (Sun, 2007). Similar experiments on other content areas (ratio, volume, and column) indicated that the findings mentioned above stand [e.g. (Wong, Lam, Sun, & Chan, 2009)].

  37. This also provide clues as to understanding the knowledge performance of the sampled Chinese teachers in the study (Ma, 1999) in the profound understanding of fundamental mathematics (PUFM) in fraction division, where a traditional textbook was used as basic channel for teaching and learning in China for over 30 years.

  38. A call for an extended comparison of examples in the exercises is recommended in different content topics across curriculum materials.

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Acknowledgments

I consider this paper as a common work that developed through the interaction of the author with the ESM editor and the reviewers whose comments were always helpful and of high level. I wish to thank Professor John Roche, university of Oxford, for his academic inspiration. I wish to thank Professor Wong, N. Y, Lam, C.C., Chinese University of Hong Kong, for the great support during my PHD study. I also wish to thank Professor lo, M. L. & Pong, W. Y., The Hong Kong Institute of Education, Marton F. & Runesson, U., University of Gothenburg, for their encouragement to study the variation approaches.

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Authors and Affiliations

  1. Faculty of Education, University of Macau, Avenida Padre Tomas Pereira S. J., Taipa, Macau, China

    Xuhua Sun

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  1. Xuhua Sun
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Correspondence to Xuhua Sun.

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This study was funded by DSEJ (Cativo.12467). A previous version of this paper appeared in Sun (2009).

Appendices

Appendix

The original examples in the Chinese textbook

  1. Example 1:

    Please calculate the given multiplication of fractions problems below, then think about what the law is and how to find the reciprocal of the numbers below: \( \frac{{3}}{{8}} \times \frac{{8}}{{3}}{,} \) \( \frac{{7}}{{{15}}} \times \frac{{{15}}}{{7}}, \) \( 5 \times \frac{{1}}{{5}}, \) \( \frac{{1}}{{{12}}} \times {12} \)

    What is the reciprocal of the numbers 1, 0?

    [Including concept of reciprocal]

  2. Example 2:

    Each box of fruit candies weighs 100 g. How much do three boxes weigh? Three boxes of fruit candies weigh 300 g. How much does each box weigh? Fruit candies weigh a total of 300 g. Each box contains 100 g of fruit candies. How many boxes are needed for these candies?

    [Including concepts of multiplication, partitive division, and equation]

  3. Example 3:

    4/5 of a sheet paper was separated into two parts. What fraction is each part of a sheet of paper? 4/5 of a sheet paper was separated into three parts. What fraction is each part of a sheet of paper?

    [Including concepts of fractions, multiplication, and partitive division]

  4. Example 4:

    Xiao Ming walked 2 km in 2/3 h, while Xiao Hong walked 5/6 km in 5/12 h. Who is faster?

    [Including partitive concepts of fraction division, and concept of fraction subtraction]

  5. Example 5:

    Xiao Hong made some flowers from an 8-meter ribbon. Each flower needs 2/3 m ribbon. How many flowers were left if she gave four of them to her classmates?

    [Including measurement concept of fraction division, partitive concept of fraction division, concept of fraction multiplication and subtraction]

  6. Example 6:

    Water makes up 2/3 of an adult's body weight and is 4/5 of a child body weight. Based on this, water in Xiao Ming’s body weighs 28 kg, and is nearly equal to the weight of water in his father's body; but Xiao Ming's body weight is 7/15 of his father's weight. What is Xiao Ming's body weight? What is Xiao Ming's father's body weight?

    [Including concept of partitive fraction division, concept of fraction multiplication, concept of equation]

  7. Example 7:

    There are 25 students in the art group, which is 1/4 more than the aero model group. How many students are there in the aero model group?

    [Including concept of partitive fraction division, concept of fraction multiplication, concept of equation]

The original examples in the US textbook

  1. Example 1:

    For an art project, Maia cuts pieces from several ribbons. How many 1/2 in. pieces can she cut from this 5-in. red ribbon? How many 3/8-in. pieces can be cut from a 3/4-in. piece? (Vol. 6, p. 290)

    [Including measure concept with fraction division algorithm]

  2. Example 2:

    A. Divide 1/2 by 1/8. B. Divide 6 by 3 C. Divide 3 by 4. (vol. 6. p. 292)

    [Including reciprocal concept, fraction division algorithm]

  3. Example 3:

    Gregory exercises by running three miles a day. Each city block he runs is 1/8 mile long. How many blocks does he run each day? How many 1/8 s are there in 1? In 2? In 3? What is 3 ÷ 1/8? (vol. 7, p. 214)

    [Including measure concept with fraction division algorithm]

  4. Example 4:

    Gramercy Park in New York City is about 2/3 of a city block long and 1/4 of a block wide. What is the area of Gramercy Park in square blocks? The city is going to subdivide six acres into lots for apartments. How many 3/4 acres lots can be made? (vol. 8, pp. 70–71)

    [Including measure concept with fraction division algorithm]

  5. Example 5:

    To multiply fractions, multiply numerators and multiply denominators.

    A. Find 1/5 × 2/3 × 1/4 B. Find 8 × 1/8 C. Find 2/3 ÷ 1/2 D. Find 3/4 ÷ 1/4

    (vol. 8. p. 584)

    [Including measure concept with fraction division algorithm]

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Sun, X. “Variation problems” and their roles in the topic of fraction division in Chinese mathematics textbook examples. Educ Stud Math 76, 65–85 (2011). https://doi.org/10.1007/s10649-010-9263-4

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