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### 數學科普 Popular Mathematics 地 點 ： 綜 合 館 H 3 0 1

TMS Annual Meeting

Speech Abstracts

### Research Institute for Math Education Tokyo University of Science E-mail: ja@jin-akiyama.com

What does it take to appreciate a musical piece? Most people must hear the music played. And if it is played by a full orchestra, then perhaps there will be excitement. What does it take to appreciate a recipe? The dish must be prepared and tasted. If it is beautifully presented and taken in the ambiance of a great restaurant, then maybe there will be excitement. The senses must be engaged. It is the same with mathematics. A mathematical concept can be exciting if it can be represented physically in a model that can be seen, manipulated, and, if possible, heard.

In this talk, we discuss mathematical models that can be used to teach standard mathematics in a non-standard way. These models can be brought into the mathematics classroom so students can work with them, experiment, discover, and gain a deep understanding of mathematical concepts. The models can be used to demonstrate the following:

1. Area and Volume 2. Pytagorean Theorem 3. Sum of Integers

4. Applications of Conic Sections 5. Figures with constant width 6. Error Correcting Code 7. Math Magic

### National Taiwan University of Science and Technology E-mail: sgshih@mail.ntust.edu.tw

SL block is an octocube that may interlock with other SL blocks to form infinite variations of stable structures. The property of self-interlocking makes SL block expressive to explore the beauty of symmetry, which has been re-garded as an essence of art and mathematics by many. This paper describes a mathematical representation that maps polynomial expressions to compositions of SL blocks. The use of polynomials, functions and hierarchical definitions simplifies the creation, communication and manipulation of complex structures by making abstractions over symmetrical parts and relationships. The discov-ery of SL block and its mathematical representation lead the way towards the development of an expressive language of forms and structures which is at the same time, rich and compact, free and disciplined.

### References

[1] Y. Estrin, A.V.Dyskin, E. Pasternak, 2011. “Topological Interlock-ing as a Material Design Concept.” Mater. Sci. Eng. C, Princi-ples and Development of Bio-Inspired Materials 31, Pages1189–1194.

doi:10.1016/j.msec.2010.11.011

[2] A.J. Kanel-Belov, A.V. Dyskin, Y. Estrin, E. Pasternak, I.A. Ivanov-Pogodaev, 2008. “Interlocking of Convex Polyhedra: Towards a Geometric Theory of Fragmented Solids.” Moscow Mathematical Journal, V10, N2, April–June 2010, Pages 337–342.

[3] S. G. Shih, “On the Hierarchical Construction of ‘SL‘ Blocks – A Genera-tive System That Builds Self-interlocking Structures.” Sigrid Adriaenssens, F. Gramazio, M. Kohler, A. Menges, and M. Pauly Eds. Advances in Archi-tectural Geometry 2016, Pages 124-137, Hochschulverlag AG an der ETH Zürich, DOI 10.3218/3778-4, ISBN 978-3-7281-3778-4

[4] P. Song, C. Fu, D. Cohen-Or, 2012. “Recursive Interlocking Puzzles.”

ACM Trans. Graph. 31 6, Article 128 (November 2012), 10 pages. DOI

### 107

= 10.1145/2366145.2366147

http://doi.acm.org/10.1145/2366145.2366147

[5] O. Tessmann, 2013. “Topological Interlocking Assemblies.” Presented at the 30th International Conference on Education and Research in Computer Aided Architectural Design in Europe (eCAADe),SEP 12-14, 2012, Czech Tech Univ, Fac Architecture, Prague, Pages 211–219.

[6] S.-Q. Xin, C.-F. Lai, C.-W. Fu, T.-T. Wong, Y. H3, and D. Cohen-Or, 2011. “Making Burr Puzzles from 3D models.” ACM Tran. on Graphics (SIGGRAPH) 30, 4. Article 97.

[7] H.T.D. Yong, 2011. “Utilisation of Topologically-interlocking Osteomorphic Blocks for Multi-purpose Civil Construction.” (PhD). University of Western Australia

### 地 點 ： 理 學 院 B 1 0 1

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TMS Annual Meeting

Speech Abstracts

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### National Central University E-mail: shann@math.ncu.edu.tw

Many grade-12 students in the course Elective Math A find Calculus actually easier to handle with than the required materials they had in grades 10 and 11.

Many math teachers agree with the students. Calculus is generally considered hard and deep since the later part of the 20th century, therefore it is isolated and reserved to the last part of the school math curriculum. We must agree that there are hard concepts in Calculus. However, those are the rigorous aspects of the subject after Cauchy, Weierstrass, and others. The naive Calculus at its early stage in the 17th century was rather intuitive and it was the source of many intriguing mathematical ideas. It is a matter of simple fact that Newton’s Principia was published only 50 years after Descartes’ La Géométrie. We all know that La Géométrie spawned Cartesian coordinates and the later fertilized Calculus and mathematical analysis. Closer historical investigations suggested that the idea of Calculus occurred to Newton and Leibniz within a decade since they were exposed to coordinate systems and early analytical geometry.

Calculus in that era was much more primitive than some materials in grades 10 and 11: space vectors, matrices and linear transformations, much of the contents in probability and statistics, to name a few.

In speaker’s opinion, to cut Calculus from the rest of math curriculum was the most unfortunate strategy made for the math education. Because calculus is the key for the sense-making math curriculum. It plays the central role that motivates and links almost all topics of school math: the very idea of rates and ratios and functions, to begin with, and it provides the reason-to-be for many topics covered in high school: the polynomial inequalities, the radian, the trigonometric and exponential functions.

In this session, the speaker will elaborate on the foregoing remarks and pro-pose a curriculum design that incorporates Calculus with the current materials naturally, organically, and painlessly.

Keywords: math education, school math curriculum, calculus

Outline