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

代数问题：美国数学教育中的代数问题越来越难解决。学生成绩的标准指标国际比较，NAEP和其他国家的措施，以及国家测试结果都指向较低的成就随着年级水平的提高。著名的TIMSS对美国课程的分析指出，基本技能建设的重复和主题覆盖的肤浅是造成这一问题的重要因素，再加上告诉和训练教学法，使学生处于过于被动的学习者地位。当然，第一年代数课的高不及格率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，但在很大程度上是无效的策略，涉及初中和高中早期课程：各种各样的预代数课程，筛选和诊断测试，地区和州强制所有学生参加代数I，延长课程，修改if代数的定义，创建不同版本的if代数，课程结束考试，等等。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特别的，而且是独立实施的，因此改进不会累积或倍增。NCTM的建议，为整合发展代数推理整个K-12只实施了少数thesstandard-based课程，这些主要是在中学水平.研究走向解决方案：几个研究小组研究早期（K-5）年级代数推理的发展，他们在本提案中有代表，已经形成了一个非正式的研究合作，延续了1990年早期/中期由美国能源部OERI资助的先前合作。这些研究人员分享了广泛的假设，关于在初等数学的背景下建立代数推理的巨大积极潜力，需要利用年轻学生中未开发的学生学习能力，概括和形式化的强大作用，加深了学生对初等数学，特别是算术的经验和理解，以及利用代数和初等数学之间课程关系的变化作为一种是小学教师专业发展的催化剂和载体。这些研究人员从事几种互补的工作，侧重于学生学习、课堂实践、课程开发、专业发展、能力建设等挑战的不同方面。所有合作研究人员都被邀请在2001年12月8日至15日在墨尔本举行的一个重要会议上发表论文，澳大利亚关于代数教学和学习的未来的报告，该报告由国际数学教学委员会赞助。教授卡普特已被指定toCo-Chair早期代数组在会议上。我们请求资金以三种方式利用这个独特的机会，为我们积极的研究计划带来强烈的国际视角：（1）支持参加国际会议的旅行。(2)在美国举行两次会议，一次在墨尔本会议之前，一次在会议之后，涉及一个稍微大一点的数学教育工作者小组。(3)撰写并出版一本专著，表达我们的研究结果，以便向政策制定者、课程开发者和评估建设者通报这一快速变化领域的最新发展。因此，我们也要求少量的资金用于编辑工作，以汇集和宣传专著，这反过来又将指向更多的网络资源，说明相关的研究。

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.