脑科学与教育】数学该怎么教？脑科学家这样说
2018-03-03 10:00北京师范大学
【脑科学与教育】是由《教育家》杂志2018年全新推出的一档学术栏目。该栏目已邀请清华大学、北京大学、北师大、东南大学等国内外高校的多位教育学家、心理学家、科学家加入。作为特约合作方，芥末堆深度参与其中，并会通过【芥末翻】栏目翻译介绍国际上以脑科学为基础的教育理念，教育技术，与《教育家》杂志一道，共同致力于推动脑科学最前沿的研究成果落地普及，让更好的教育来得更快！
图片来源：摄图网
数学教育界一直存在着两种不同的声音，一种注重数学的学习过程和价值，使数学回归生活中，在具体的情境中学习数学，强调数学的“生活化”“生活味”。另一种则是强调从数学学科体系的角度出发，注重构建较为严密的数学学科知识体系，突出“数学化”“数学味”。这两种趋势我们概括为情境化和符号化。面对数学教育中“符号化”与“情境化”这一历久弥新的争论，总体上，具有教育学、心理学背景的专家学者更倾向于“生活化”即情境化数学；而有数学专业背景的专家学者则倾向于“数学化”即符号化数学；但也已有学者认识到二者必须相互融合，并提出了对应的方法。随着脑与认知科学的兴起，对数学认知的脑机制研究不仅为数学化与生活化的融合需要提供了有力的脑科学证据，而且为二者的融合提出了新的方法与启示。
情境化与符号化融合的认知与脑机制
图片来源：pixabay
数学“生活化”即情境化是在具体的生活情境中解决数学问题，与推理、言语理解及情境记忆等都有密切的关系，其脑机制与额叶、颞叶、海马等区域有关。研究发现比起纯算式的问题，学生更容易回答有情境的数学问题，如0.3×40，和“一根铅笔0.3元，我要买40根应该付多少钱？”。而且问题陈述更为详细时解决起来更容易。再比如，同样的数字“1620”“1789”，作为数字比较大小和作为情境事件的年份比较时，共享顶叶激活，但年份比较时负责语义加工的颞叶和额叶有更多激活。脑成像研究进一步发现应用题的推理加工成分激活了顶叶，对题目中言语的理解激活了颞叶和额叶。另外在处理新情境中的数学问题时，会涉及到从情景记忆中提取有用的线索信息，其脑机制和内侧颞叶、海马等脑区相关。
数学教育的“数学化”即符号化内容则包括了数字数量加工、数学计算、符号加工推理等内容，其脑机制主要定位为大脑顶叶区域。研究中发现不论阿拉伯数字、中文、英文、罗马数字等形式，包含数字数量加工的条件引起了更多顶叶区域激活。如进行判断“2，5，8，11，下一个是？”之类的数字归纳推理任务时顶叶区域会被显著激活。数学认知的“形状加工假设”指出数学符号相关的数学认知都是基于符号的形象加工的，顶叶则在不同类型数学任务中发挥着这些形象的空间加工的作用。
数学活动是多个脑区的共同作用，已有研究发现了一些脑区在数学认知中具有较强的功能连接。例如额叶、海马等的功能连接强度可以预测儿童的数学学习效果；额叶和顶叶间的功能连接在计算加工中起到重要作用。而笔者最新研究显示精确计算时顶叶、中央沟区域和海马三者之间的连接也是显著的。脑功能的连接，证明了数学教学中，情境与符号的融合是符合大脑活动规律及数学认知机制的。
言语对二者融合的调和
不论是数学应用题等情境化内容还是数字符号计算、推理等符号化内容的加工，都一定程度的激活了大脑言语加工脑区。首先，情境编码和语义编码时额叶、颞叶等的大脑激活模式在很大程度上是相同的，且右侧额叶和海马区域的激活程度可以某种程度上预测记忆的效果。因此，情境和语言具有一定的共通成分，可以从情境中提取语言内容，也可以用语言激活情境记忆。
其次，研究发现数学术语的加工主要依赖于语言音形义三要素中的语义脑区，即定位于颞叶与额叶；类似加法交换律等算术原理的加工除了顶叶激活，负责语言加工的颞叶区域也有显著激活；针对应用题和几何证明题的数学问题解决也依赖于大脑语义网络，并且左半球的角回、颞叶、额叶等都有明显激活。
可见言语加工是数学加工中的重要成分，情境数学与符号数学均需要语义网络的参与作用。因此，情境与符号的融合还能以语言为中介。结合情境、符号和语言在数学加工中的独特参与作用，在实践中需要实行三元数学教育。
三元数学教育
图片来源：教育家
三元数学包含情境数学、符号数学和言语数学三部分。其中的情境数学指包含数学原理的情境。符号数学是采用特定的数学符号，如数字和字母等，对数学原理加以表示。言语数学主要指用自然语言对数学原理进行描述。例如，对于数学原理“加法交换律”，具体情境即为“两个篮子的鸡蛋交换后总数不变”，用语言描述其蕴含的数学原理为“两个数相加，交换位置和不变”，符号数学可以表达为“a+b=b+a”。三者之间可以相互转化、相互补充以促进数学理解和记忆。
世界及我国教育改革所面临的“度”的问题，既有过于侧重情境化造成的数学知识体系建构不完全；亦有过于侧重符号化而造成的学生理解困难。三元数学中，第一个问题可以通过语言描述的数学知识体系框架来解决；第二个问题则可以言语为中介理解抽象的数学符号，同时对符号背景、历史、发展的言语描述也能建立起有情境支撑的符号系统。这主要包含四个层面。
第一，教学的起点可以是情境，也可以是符号。我国新课改以来的新教学过程即首先创设教学情境，将具体情境中的数学原理转化为言语描述，再进一步转化为数学符号表达。另一种即从符号入手教学，可以使用言语描述符号的历史、读写法、含义，进而涉及具体情境中的应用。两种思路殊途同归。
第二，情境、语言和符号不构成递进关系，而是相互转化的关系。三者转化的过程是可逆的：既可以用言语描述情境，亦能将描述数学原理的言语转化为符号。言语与符号加工共享的大脑额叶、颞叶等区域为此提供了脑基础，而且研究显示这些区域的功能连接是双向的，并不是单向的。因此，言语数学的应用及对情境与符号的融合是一个相互的关系，而不是单方面的递进。
第三，在情境、语言和符号中以理解和应用数学原理为中心。数学原理包括数学定理、公式、法则等，是对概念的属性以及概念之间关系的逻辑判断。数学原理是数学教学的中心，因其不仅是数学情境的核心，也是数学符号的基础。而算术原理的加工依赖于和数学符号相关的顶叶区域、和语言相关的颞叶、额叶区域等，这说明了对数学原理的学习需要三者的共同作用。
第四，数学教育目标的界定不是从情境到符号，而是言语、符号到情境及其转换的多重目标；对应的评价也要从情境、言语和符号三个维度考虑。我国数学教育很长一段时间内都是以符号数学为主要目标及评价标准的；新课改后增加了能力目标，教师学生普遍开始接受以在具体情境中解决问题的能力作为教育目标。目前人们发现情境化数学导致数学成绩下降，可能与评价仍然是符号化数学的评价有关，而增加以数学的理解能力为目标的评价是解决方法之一。如考试中适当增加数学原理理解题的比例，考察对数学原理这一核心目标的理解。
作者介绍：
周新林，北京师范大学认知神经科学与学习国家重点实验室教授，博导；北京师范大学Siegler创新学习中心主任（中方）；国际数学认知与学习协会(iMCLS)理事。
（原文刊于《教育家》2018年一月刊，本文略有删减）返回搜狐，查看更多
声明：本文由入驻搜狐号的作者撰写，除搜狐官方账号外，观点仅代表作者本人，不代表搜狐立场。
とても興味深く読みました：ゼロ除算の発見4周年超えました：
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 412: The 4th birthday of the division by zero $z/0=0$ \\
(2018.2.2)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).
In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.
For the details, see the references and the site: http://okmr.yamatoblog.net/
We wrote a global book manuscript \cite{s18} with 154 pages
and stated in the preface and last section of the manuscript as follows:
\bigskip
{\bf Preface}
\medskip
The division by zero has a long and mysterious story over the world (see, for example, H. G. Romig \cite{romig} and Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta (598 -668 ?) established the four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in
Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is right and suitable.
The division by zero $1/0=0/0=z/0$ itself will be quite clear and trivial with several natural extensions of the fractions against the mysterously long history, as we can see from the concepts of the Moore-Penrose generalized inverses or the Tikhonov regularization method to the fundamental equation $az=b$, whose solution leads to the definition $z =b/a$.
However, the result (definition) will show that
for the elementary mapping
\begin{equation}
W = \frac{1}{z},
\end{equation}
the image of $z=0$ is $W=0$ ({\bf should be defined from the form}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere (\cite{ahlfors}). �As the representation of the point at infinity of the Riemann sphere by the
zero $z = 0$, we will see some delicate relations between $0$ and $\infty$ which show a strong
discontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function $W =1/ z $ at the origin $z = 0$, because we did not consider the division by zero
$1/ 0$ in a good way. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or the
point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or
based on the basic idea of Aristotle. --
For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we will consider its value of
the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in
mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential equations, complex analysis in the undergraduate level and to our basic ideas for the space and universe.
We have to arrange globally our modern mathematics in our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and the universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is in the undergraduate level.
\bigskip
\bigskip
{\bf Conclusion}
\medskip
Apparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.
This book is an elementary mathematics on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.
The division by zero theory may be developed and expanded greatly as in the author's conjecture whose break theory was recently given surprisingly and deeply by Professor Qi'an Guan \cite{guan} since 30 years proposed in \cite{s88} (the original is in \cite {s79}).
We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.
We have to change our basic ideas for our space and world.
We have to change globally our textbooks and scientific books on the division by zero.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{guan}
Q. Guan, A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms16}
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
\bibitem{ms18}
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Submitted for publication)
\bibitem{mms18}
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{msy}
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 2017), 1-16.
\bibitem{osm}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.
\bibitem{os}
H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).
\bibitem{o}
H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.
\bibitem{os18}
H. Okumura and S. Saitoh,
Applications of the division by zero calculus to Wasan geometry.
(Submitted for publication).
\bibitem{ps18}
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{romig}
H. G. Romig, Discussions: Early History of Division by Zero,
American Mathematical Monthly, Vol. {\bf 3}1, No. 8. (Oct., 1924), pp. 387-389.
\bibitem{s79}
S. Saitoh, The Bergman norm and the Szeg$\ddot{o}$ norm, Trans. Amer. Math. Soc. {\bf 249} (1979), no. 2, 261--279.
\bibitem{s88}
S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 189}. Longman Scientific \& Technical, Harlow; copublished in the United States with John Wiley \& Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3
\bibitem{s14}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
\bibitem{s17}
S. Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM](2017.12.17).
\bibitem{s18}
S. Saitoh, Division by zero calculus (154 pages: draft): （http://okmr.yamatoblog.net/）
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{a}
https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf
\bibitem{b}
http://publish.uwo.ca/~jbell/The 20Continuous.pdf
\bibitem{c}
http://www.mathpages.com/home/kmath526/kmath526.htm
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.
\bibitem{ann354}
Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.
\bibitem{362}
Announcement 362(2017.5.5): Discovery of the division by zero as $0/0=1/0=z/0=0$
\bibitem{380}
Announcement 380 (2017.8.21): What is the zero?
\bibitem{388}
Announcement 388(2017.10.29): Information and ideas on zero and division by zero (a project).
\bibitem{409}
Announcement 409 (2018.1.29.): Various Publication Projects on the Division by Zero.
\bibitem{410}
Announcement 410 (2018.1 30.): What is mathematics? -- beyond logic; for great challengers on the division by zero.
\end{thebibliography}
\end{document}
List of division by zero:
\bibitem{os18}
H. Okumura and S. Saitoh,
Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum.
Saburou Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM]
arXiv:1712.09467 [math.GM]
Hiroshi Okumura and Saburou Saitoh
The Descartes circles theorem and division by zero calculus. ２０１７．１１．１４
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Submitted for publication).
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
再生核研究所声明371（2017.6.27）ゼロ除算の講演― 国際会議 https://sites.google.com/site/sandrapinelas/icddea-2017 報告
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
ソクラテス・プラトン・アリストテレス その他
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか
〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか
NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか
再生核研究所声明 411(2018.02.02): ゼロ除算発見4周年を迎えて
ゼロ除算の論文
Mysterious Properties of the Point at Infinity
Mysterious Properties of the Point at Infinity
Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197
0 件のコメント:
コメントを投稿