数学建模教学研究的 国际动态与思考 第13届全国数学建模教学与应用会议(2013, 西安) 谢金星 Department of Mathematical Sciences Tsinghua University, Beijing 100084, China Tel.: 010-6278 7812 E-mail: jxie@math.tsinghua.edu.cn

内容提要 ICME-12 (2012, South Korea) 简介 ---- 与数学建模有关的一些内容 2. ICTMA-16 (2013, Brazil) 介绍 ---- 大会报告 ---- 分组报告等 3. 中外对比与思考 ---- 借鉴与改进 ---- 国际化

http://www.mathunion.org/icmi ICME: Intl. Congress on Mathematical Education ICME-1: 1969, France (Hans Freudenthal倡导) ICME-2: 1972, UK (此后每4年一届) Organized by ICMI (founded in 1908): Intl. Commission on Mathematics Instruction, a commission of the International Mathematical Union (IMU) starting from 1952. 宗旨：展示数学教育研究的现状和趋势，以及 所有层次上的数学教育研究和实践

84国、3616人 韩国1641人 中国大陆282人， 位列第二， 中学教师为主 日本232人， 位列第三 http://icme12.org 数学教育在东亚 (PA-6, Panel)

ICME-12 (http://icme12.org) ICMI设立的两个奖项： Hans Freudenthal奖 Felix Klein奖 （都是每两年评奖一次，本次会议上授予的是2009年和2011年的四位获奖者）。 数学嘉年华（MC - Mathematical Carnival）活动： 与大会全程相伴，展示数学教育书籍、技术器材、软件、教具、玩具、游戏等

WSG-4：Math Communication & Use of Models to Solve Problems ICME-12 Program 8 PA-8: Quality teaching of math modeling: What do we know, What can we do? 5 ST-4： Key mathematical concepts in the transition from secondary to university TSG-17: Math applications and modelling in the teaching and learning of math 37 DG-14：Math Modeling in Connecting Concepts to Real World Application 14 41 WSG-4：Math Communication & Use of Models to Solve Problems ICTMA 5 EIMI

EIMI: The 20th ICMI Study The 14th ICMI Study Blum, W., Galbraith, P., Henn, H.-W., & Niss, M. (Eds.) 2007, 524p. 是建模教育的重要参考书 The 20th ICMI Study (Jointly with ICIAM) Educational Interfaces between Mathematics and Industry (EIMI) 即将出版

Quality teaching of math modeling: What do we know, What can we do? 报告人：Werner Blum, Univ. of Kassel, Germany Two introductory real world examples Mathematical modelling competency Students’ modelling activities Aims and perspectives of modelling Teaching modelling Teacher support for modelling activities Strategies for learning modelling Teacher competencies for modelling A final real world example

Effective/quanlity teaching of a&m (applications & modelling) What do we know empirically about effective teaching of a&m? Generally speaking, well-known findings on quality teaching of mathematics hold, of course, also for teaching of mathematics in the context of relations to the real world 10 important aspects for teaching methodology for a&m:

Quality teaching of a&m 1 Necessary: effective and learner-oriented classroom management (using time effectively, separating learning and assessment recognisably, using students’ mistakes constructively, varying methods and media flexibly, ... Hattie&Timperley 2009; Helmke&Weinert, Baumert et al., Klieme) Group work particularly suitable for modelling (Ikeda&Stephens) Co-constructive group work (Reusser et al., Messner et al.)

Quality teaching of a&m 2 Necessary: activate learners cognitively, stimulate students’ own activities (“Modelling is not a spectator sport”, Schoenfeld et al.) Distinguish carefully between students working independently with teacher support and students working alone Crucial: permanent balance between students‘ independence and teacher‘s guidance (“Principle of minimal support” Aebli)

Quality teaching of a&m 3 Necessary: activate learners meta-cognitively (accompanying and retrospective reflections, advancing strategies)

Quality teaching of a&m 4 Necessary: variety of suitable examples (transfer cannot be expected); in particular: variation of - real world contexts (Busse) - mathematical contexts and topics Authenticity not always required but helps to reduce “suspension of sense-making” substantially (Palm, Verschaffel et al.)

Quality teaching of a&m 5 Teachers ought to encourage individual solutions In practice, teachers tend to favour their own solution (Leikin, DISUM) Reasons for encouraging multiple solutions (Hiebert&Carpenter, Rittle-Johnson&Star, Schoenfeld, Krainer, Stiegler&Hiebert, Tsamir et al., Neubrand) MultiMa (Schukajlow/Krug): Students who developed several solutions have higher learning gains

Quality teaching of a&m 6 Competencies evolve in long-term learning processes, beginning with early “implicit models” (Lesh& Borromeo Ferri, Greer&Verschaffel) Necessary: repeating and practising Balance between focus on sub-competencies of modelling and focus on modelling competency? Needed: competence development model(s) Approach to describe competency development ~ KOM (Niss, Blomhøj, Jensen)

Quality teaching of a&m 7 Also assessment has to reflect the aims of a&m appropriately; analogous quality criteria (Berry, Haines&Crouch, Houston&Neill, Burkhardt, Izard et al., Antonius/Niss et al., Galbraith&Stillman, Vos, DISUM) Functions of tests: norms, illustration, diagnosis Model (Zöttl/Ufer/Reiss 2011) for assessing modelling sub-competencies and general modelling competency

Quality teaching of a&m 8 Parallel development of competencies and beliefs & attitudes →long-term learning processes

Quality teaching of a&m 9 Digital technologies can be powerful tools: experiments, investigations, simulations, visualisations or calculations (Hoyles&Noss, Galbraith&Geiger, Henn, Wake, Borba, Lingefjärd, Greefrath&Siller, ...) Extending the modelling cycle (Greefrath) Needed: more studies into the effects of technologies

Quality teaching of a&m 10 Case studies: Mathematical modelling can be learned by students supposed there is quality teaching (Abrantes, Kaiser, Galbraith&Clatworthy, Maaß, Biccard&Wessels, DISUM,...) However: More research needed, especially small scale studies using a mixture of qualitative and quantitative methods

Quality teaching of a&m All these activities will not be sufficient Implementation of applications and modelling has to take place systemically:

Teacher competencies for modelling Teacher’s PCK  students’ achievement gains

Teacher competencies for modelling PCK (Ball&Bass, TEDS-M, COACTIV; Maaß, Doerr, Lingefjärd, Kaiser/Schwarz/Buchholtz; Ferri &Blum)

内容提要 ICME-12 (2012, South Korea) 简介 ---- 与数学建模有关的一些内容 2. ICTMA-16 (2013, Brazil) 介绍 ---- 大会报告 ---- 分组报告等 3. 中外对比与思考 ---- 借鉴与改进 ---- 国际化

ICTMA有两层意思 1. International Community of the Teachers of Mathematical Modelling and Applications ----ASG (Affiliated Study Group) of ICMI (Intl. Commission on Mathematics Instruction): International Study Group for the Teaching of Mathematical Modelling and Applications 2. International Conference on the Teaching of Mathematical Modelling and Applications ---- 1983开始举办，每两年一届 网址: www.ictma.net

ICTMA-16: Blumenau, Brazil 近40小时的旅程！

ICTMA-16基本情况 约30个国家，约150人，所有洲都有，巴西最多； 其他国家包括：德国、澳大利亚、美国、日本、 中国、荷兰、南非、墨西哥、阿根廷、英国、… 中国内地5人(国防科大、清华、华东理工）

Mathematical Modeling: history and future perspectives ICTMA-16基本情况 报到7月14日；会议14日晚（开幕）-19日中午； 14日上下午执委会；19日上午后半段Business Meeting 一次旅游安排（半天：海滩/公园） 　５个大会报告(报告１小时，评述/提问30分钟) 　81个分组报告（40分钟，3－５组同时进行） 　1次“Panel Discussion”(１.５小时): 主题研讨 巴西特色：巴西专场(Poster)、Math Fair 会议主题： Mathematical Modeling: history and future perspectives

5个大会报告 Henry Pollak - Columbia University, United States THE PLACE OF MATHEMATICAL MODELING IN THE SYSTEM OF MATHEMATICS EDUCATION: PERSPECTIVE AND PROSPECT (数学建模在数学教育体系中的位置：透视与展望) Mongens Niss – Roskilde University (RUC), Denmark PRESCRIPTIVE MODELLING - OPPORTUNITIES AND CHALLENGES (规范/解释/说明型模型-机会与挑战) Gabriele Kaiser – University of Hamburg, Germany MODELLING COMPETENCIES - IMPORTANCE IN THE PAST DEBATE AND FURTHER PERSPECTIVES (建模能力 – 在过去争论中的重要性及未来展望)

5个大会报告 Ubiratan D’Ambrosio – UNIBAN, BRAZIL MATHEMATICAL MODELLING AS A STRATEGY FOR BUILDING-UP SYSTEMS OF KNOWLEDGE IN DIFFERENT CULTUREAL ENVIRONMENTS (数学建模作为不同文化环境下知识系统的建立策略) Rodney C. Bassanezi – UFABC-CMCC, BRAZIL 　 TEACHING AND LEARNING WITH MATHEMATICAL MODELLING – TRAINING COURSES (通过数学建模培训课程进行教与学)

Pollak: PLACE OF MM MATHEMATICAL MODELING: A NEW PRODUCT IN A COMPLEX SYSTEM  How does Mathematical Modeling Affect: (A) The Relationship Between Secondary and Tertiary Education Modeling in the high schools – what will universities do? (US) Modeling in the universities -- what will high schools do? (China)

Pollak: PLACE OF MM (B) Teacher Educators preparing teachers to teach mathematical modeling (1) Teaching about the modeling process. Can it involve new mathematics? (2) Assessment. How do you judge the success of a model? (3) What do teachers believe about (from a survey) (a) Mathematical Models (b) Mathematical Modeling (c) The Teaching of Modeling (4) The history of my personal relation to modeling (5) A Modeling Problem （车速估计）

Pollak: PLACE OF MM (C) The Relations between mathematics and mathematics education   (1) Modeling as Vehicle versus Modeling as Content. (2) Primary School (3) Secondary School (4) Back to (A1) : The Secondary-Tertiary Interface (5) Modeling as a source of new insights on mathematics itself

Extra-mathematical domain D Niss: PRESCRIPTIVE MODELLING 相对于descriptive modelling: 建模过程有何相同、不同？ The modelling cycle Extra-mathematical domain D Speci- fication Ideali- sation Idealised situation cum questions Mathematical domain M Mathematised situation cum questions Mathemat-ical answers de-mathematisation Mathematisation f translation answers interpretation 如： BMI、 GINI、 … 检验： 不可证伪 容易批评

Challenges and opportunities We have to devise teaching and learning activities for prescriptive modelling, also for situations in which students cannot complete the rudimentary modelling cycle unassisted. Two focal points of such activities (opportunities): Students should be engaged in analysing the hidden or explicit assumptions, requirements, wishes and pre-requisites underlying the mathematisation (to be) undertaken. Students should be engaged in meta-validation of models

Kaiser: Modelling Competencies Important theoretical and empirical approaches with a special focus on the relation of modelling competencies and modelling process Important approaches for measuring modelling competencies (including sub-competencies ) Own activities to promote modelling competencies Modelling days / modelling weeks Effectiveness of the holistic versus the atomistic approach to foster modelling competencies

D’Ambrosio: MM as a Strategy … CENTRAL QUESTION: How do humans develop strategies which are the basic components of the complex systems of knowledge and behavior?  ETHNOMATHEMATICS (D'Ambrosio in 1977) ↔ TICS of MATHEMA in different ETHNOS 民族数学？ 民俗数学？ 民俗科学？

D’Ambrosio: MM as a Strategy … The full cycle of knowledge math modelling (not easy!)

Bassanezi : MM Training Courses Modelling movement and its development in Brazil Modelling courses and modelling movement Teacher’s barriers to the implementation of modelling New trends in mathematical topics (including technology), e.g. big data

分组会形式 3－５组同时进行 -- 每人报告25-30分钟； -- 讨论（提问／回答）15-20分钟; 　3－５组同时进行 -- 每人报告25-30分钟； -- 讨论（提问／回答）15-20分钟; -- “教学”为主（中小学为主），“应用”较少 充分交流 这是这个系列会议的一个传统和特点

中国与会者的报告（大学层次） 吴孟达:　 THE TEACHING GOAL AND ORIENTED LEARNING OF MATHEMATICAL MODELING COURSE 段晓君： 国防科大的数学建模活动 + 卫星定位问题建模 鲁习文： 华东理工的数学建模活动 + 在线调度问题建模 谢金星： 　数学建模夏令营（大学）

Math Fair

内容提要 ICME-12 (2012, South Korea) 简介 ---- 与数学建模有关的一些内容 2. ICTMA-16 (2013, Brazil) 介绍 ---- 大会报告 ---- 分组报告等 3. 中外对比与思考 ---- 借鉴与改进 ---- 国际化

一点感受：中外对比 外：多数人来自大学的数学教育系（教师/研究生） -- 不太关注大学数学，更关注中小学数学教育 中：非教育系、大学数学教育为主 外：多从教育学、心理学、认知科学等角度研究 -- 注重定量研究，或定量与定性研究相结合 -- 注重方法论研究(规范的研究方法) -- 注重实证研究 (实验对比；问题要具体) 中：定性、经验为主 森林树木？ 中医西医？

一点感受：借鉴与改进 研究问题越具体越好，不求大而全 切忌写成年终报告、成绩总结、…… （领导保障，教师关键，学生主体  课程设置与教学，竞赛组织和成绩） 更深入细致、规范的教学研究，例如： 如何衡量某种教学法比另一种好？（指标、实验） 　 学生建模时难点在哪？原因何在？（认知过程） 如何设计教学案例/过程？教师作用？（教学法）

一点感受：教学案例 对于建模应用方面的研究（实际建模问题） -- 注重问题来源是否真的实际 -- 注重文献综述（目前相关研究的现状） 　对于建模应用方面的研究（实际建模问题） -- 注重问题来源是否真的实际 -- 注重文献综述（目前相关研究的现状） -- 注重结果的分析／检验 -- 注重实际解决问题的效果 -- 注重能否在课堂教学中使用、如何使用的建议 （EIMI: 数学与工业的教育交融） 期待中国在这方面有更多的研究成果！

一些期刊 数学建模及其应用 数学的实践与认识 中国大学教学 中国高教研究 高等理科教育 职业教育研究 中等职业教育 …… Teaching Mathematics and its Applications ZDM; J. Math Didakt J. of Math. Modeling and Application Educational Studies in Mathematics Research in Mathematics Education Int. J. of Computers for Math. Learning Int. J. of Math. Education in Sci. & Tech. Int. J. of Science and Math. Education Journal of Mathematical Behavior …… 数学建模及其应用 数学的实践与认识 中国大学教学 中国高教研究 高等理科教育 职业教育研究 中等职业教育 ……

ICTMA后几届会议 ICTMA-17 (2015): Centre for Research in Mathematics Education (CRME), University of Nottingham, U.K. ICTMA-18 (2017): 待2015年ICTMA-17期间确定 (如申办，应有所准备) 尽量多参加会议； 尽量使研究规范； 尽量体现“教学”。 与会者和执委会对于在中国举办有兴趣，但也有疑虑

会议论文集 谢