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

TMS Annual Meeting

### 數 學 年 會 2018 ^{數 學 年 會}

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### 演講摘要

Speech Abstracts

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### What Does it Take to Get Ichiro and Hanako Excited About Mathematics?

### Jin Akiyama

### 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

### 106

*The Art and Mathematics of Self-interlocking SL* Blocks

### Shen-Guan Shih Department of Architecture

### 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

### 108

### Mathematics Education 數學教育

### 地 點 ： 理 學 院 B 1 0 1

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TMS Annual Meeting

### 數 學 年 會 2018 ^{數 學 年 會}

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### 演講摘要

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### 109

### 微積分了沒？

### 微積分教材嚴格化的反思以及對高中和大一 課程的啟示

### 單維彰 Wei-Chang Shann Department of Mathematics

### 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**

### 110

### 人工智慧與學習分析在數學教育上的應用

### 郭伯臣

### 教育資訊與測驗統計研究所 國立臺中教育大學 E-mail: kbc@mail.ntcu.edu.tw

台灣教育部於 2017 年啟動了一個適性學習平台「因材網」來協助教師進行 適性教學與學生適性學習，在因材網中有國小與國中數學學習內容，包含教學 影片、練習題、動態評量、對話式智慧教學系統，本演講將以因材網中的數學 數位學習內容與工具為基礎，說明人工智慧與學習分析在數學學習上的應用，

包含：一、數學智慧教學系統設計與成效；二、學習分析模式與在因材網數學 學習資料分析；三、因材網應用在數學補救教學方式與成效。