Exploring Liu Hui’s Cube Puzzle - Conclusion and Acknowledgments

Conclusion

Liu Hui's Cube Puzzle has proven a worthwhile case for teaching and learning of mathematics using historically significant tasks. It is well suited for K-12 and teacher education math classes as an art project, a geometric construction project, or a pedagogical task for teachers to reflect on the art of mathematics teaching and learning. In the lower grades, the focus can be on mathematical literacy in order for children to develop the vocabulary and visual experience for spatial reasoning. In the middle grades and above, the focus can be on geometric construction, geometric proof, 3-D design, and even Liu Hui's infinitesimal argument. Perhaps, the goal is not really making or solving the cube puzzle; rather, it is a way of thinking about the mathematical relevance of all kinds of cultural tools or designs in the history of mathematics, science, and culture. With the daily advancement of digital technologies, mathematical treasures of the past, whether recreational or professional, have acquired a new status as objects of mathematical teaching and pedagogical innovation. Initially, I had a hard time finding the historical source of the puzzle, due in part to the foreignness of the original Chinese terms. Once I located the historical origin of the cube puzzle, I found an excitingly rich world of mathematics, much of which I had heard about before, but for which I now have fresh connotations from a contemporary perspective, particularly in terms of K-12 mathematics teacher education. During my historical excursion, I encountered Friedrich Froebel, who invented similar puzzles and blocks in Europe more than 150 years ago (Kriege, 1876). I have also read Martin Gardner (2001), Brian Bolt (1993), H. Steinhaus (1969), Ian Stewart (2009), and H. Martyn Cundy and A. P. Rollett (1961), much of whose recreational mathematics is serious mathematics under a playful cover and invites fresh visits, with either traditional or contemporary technologies.

In the case of cube dissection, Liu Hui may not have intended his method to become a puzzle, as can be seen from his clear focus on practical matters of his own time. Nevertheless, it has over the centuries evolved into one of the most engaging mathematical inventions, frequently covered in texts on the history of mathematics, in China and internationally. From a modern perspective, Liu Hui's Cube Puzzle is also symbolic of a way of thinking about the mathematics education of school students and their teachers, an invitation to reconsider the design metaphor for mathematics teaching and learning, for, after all, our knowledge is a kind of design (Perkins, 1986). It is equally interesting as a case that supports the integration of mathematical history in mathematics education, an area that has attracted both mathematicians and mathematics educators in recent years. There is a wealth of similar projects, including much of Liu Hui's 1800-year-old commentary, that lend themselves to school students and mathematics teachers. When classic problems converge with modern technologies, mathematics educators have a fresh opportunity to engage children and adults in exploring and appreciating the nature and power of mathematical thinking.

Acknowledgments

The author is grateful to the anonymous reviewers and the MAA Convergence Editor, Dr. Janet Beery, for their constructive suggestions and thought-provoking questions, without which the article would not have come to its present shape. In addition to her kind encouragement, Dr. Beery took the trouble of locating some key references, which led the author to a rewarding journey of reading and research in ancient Chinese texts, modern Chinese interpretations, and English translations in the context of the problem. Special thanks are due to all the prospective and classroom teachers for their enthusiastic participation in the author's mathematics teaching experiments. The author is solely responsible for any mistakes or misinterpretations.