An International Perspective on the Role of Algebra in Elementary School Mathematics

代数问题：美国数学教育面临着越来越难解决的代数问题。学生成绩的国际比较标准指标、净资产收益率和其他国家指标，以及国家测试结果，都指向随着年级水平的提高，学生成绩下降的趋势。众所周知，TIMSS对美国课程的分析指出，基础技能培养的重复和主题覆盖的肤浅是导致这一问题的重要原因，再加上将学生置于过于被动的学习者地位的告知和演练教学法。当然，一年级代数课程的高失败率也导致了学生的疏远和教师的士气低落。应对措施涉及各种各样的策略，但很大程度上是无效的，涉及初中和早期的课程：各种代数预科课程、筛查和诊断测试、地区和州要求所有学生参加代数I、延长课程、修改IF代数的定义、创建不同版本的IF代数、课程期末考试等。每一次调整都会产生微小的改进，尽管由于这些策略往往是临时的，并独立实施，因此改进不会累积或成倍增加。NCTM关于整合整个K-12年级代数推理发展的建议只在少数基于标准的课程中实施，这些建议主要在中学水平上实施。解决方案研究：几个研究早期(K-5)年级代数推理发展的研究小组，他们在这个建议中代表，组成了一个非正式的研究合作，更新了美国能源部OERI在1990年初/中期资助的先前的合作。这些研究人员分享了广泛的假设，包括在小学数学背景下建立代数推理的巨大积极潜力，在年轻学生中开发尚未开发的学生学习能力的必要性，概括和形式化提高学生对初等数学特别是算术的体验和理解的强大作用，以及利用代数和初等数学之间课程关系的变化作为小学教师教师发展的催化剂和工具的潜力。这些研究人员从事着几个互补的工作，专注于学生学习、课堂实践、课程开发、专业发展、能力建设等不同方面的挑战。旅行、会议和专著的建议组合：合作的研究人员都被邀请在2001年12月8-15日在澳大利亚墨尔本举行的关于代数教学和学习的未来的重要会议上发表论文，该会议由国际数学教学委员会赞助。卡普特教授在会议上被任命为早期代数小组的主席。我们请求资金，通过三种方式利用这一独特的机会，将强烈的国际视角应用到我们积极的研究计划中：(1)支持参加国际会议的旅行。(2)在美国举行两次会议，一次是在墨尔本会议之前，一次是在墨尔本会议之后，涉及这一领域的数学教育工作者的人数略多一些。(3)撰写和出版一本专著，表达我们的发现，以便向政策制定者、课程制定者和评估建造者通报这一快速变化领域的最新发展。因此，我们还要求为编辑工作提供少量资金，以整合和宣传专著，这反过来将指向说明相关研究的额外网络资源。

The Algebra Problem: US Mathematics Education has an increasingly difficult in AlgebraProblems to solve. The standard indicators of student achievements international comparisons, NAEPand other national measures, as well as state test results all point in the direction of lower achievementas grade levels increase. Well-known TIMSS analyses of US curricula point in the direction of repetitionof elementary skill building and superficiality of topic coverage as important contributors to the problem,combined with tell-and-drill pedagogy that puts students in overly passive position as learners. And, ofcourse, high failure rates in first year algebra courses contribute to alienation on the part of students andlow morale among teachers. Responses involve a wide variety, but largely ineffective tactics involvingmiddle and early high school curricula: pre-algebra courses of various kinds, screening and diagnostictests, district and state mandates that all students take Algebra I, lengthened courses, modifying thedefinition of if algebra, creating different versions of if algebra, end-of-course exams, among others.Each adjustment produces small improvements, although, since the tactics tend to be ad hoc andimplemented independently, improvements do not accumulate or multiply. NCTM recommendations forintegrating the development of algebraic reasoning across K-12 are implemented only in a few of thestandards-based curricula, and these mainly at the middle school level.Research Towards Solutions: Several research teams studying the development of algebraicreasoning in the early (K-5) grades, who are represented in this proposal, have formed an informalresearch collaborative, renewing a prior collaboration funded by the US DoE OERI in the early/mid1990. These researchers share broad hypotheses regarding the large positive potential of buildingalgebraic reasoning in the context of elementary mathematics, the need to exploit untapped studentlearning capacities among younger students, the powerful role of generalization and formalization indeepening students experience and understanding of elementary mathematics especially arithmeticand the potential for using changes in the curricular relationships between algebra and elementarymathematics as a catalyst and vehicle for faculty development of elementary teachers. These researchersare engaged in several complementary lines of work focusing on different aspects of the challengesstudent learning, classroom practice, curriculum development, professional development, capacitybuilding, and so on.Proposed Combination of Travel, Conferences and Monograph: The collaborating researchershave all been invited to present papers to an important conference to be held December 8-15, 2001, inMelbourne, Australia on The Future of the Teaching and Learning of Algebra which is being sponsoredby the International Commission on Mathematics Instruction. Professor Kaput has been designated toCo-Chair the Early Algebra Group at the conference. We request funds to exploit, in 3 ways, this uniqueopportunity to bring an intensely international perspective to bear upon our active research program: (1)To support travel to the international conference. (2) To hold two meetings in the US, one prior to andone following the Melbourne conference, involving a slightly larger group of mathematics educatorsworking in this area. (3) To write and publish a monograph expressing our findings in order to informpolicy makers, curriculum developers and assessment builders of the latest developments in this rapidlychanging field. Hence we also request a small amount of funding for editorial work to pull together andpublicize the monograph, which in turn will point to additional web resources illustrating relevantresearch.