Conceptual Model-based Problem Solving: A Response to Intervention Program for Students with Learning Difficulties in Mathematics

探索研究K-12计划(DRK-12)旨在通过研究和开发创新资源、模型和工具，显著提高K-12学龄前学生和教师在科学、技术、工程和数学(STEM)方面的学习和教学。DRK-12计划中的项目建立在STEM教育的基础研究和先前的研究和开发工作的基础上，为拟议的项目提供了理论和实证证明。为期3年的探索性项目-基于概念模型的问题解决：对数学学习困难学生干预计划的响应，将开发一个跨平台的数学辅导计划，以解决数学学习困难学生(LDM)二年级和三年级学生解决问题的技能困难。虽然数学问题解决技能在日常生活的各个领域都是至关重要的，但许多学习困难的学生在上学初期并没有熟练地掌握关键的数学概念，如加法和乘法推理。事实上，大约5%-10%的学龄儿童被认定患有数学残疾，这可能会导致他们在高年级经历相当大的困难，并经历持续不断的学业、生活和工作挑战。尽管针对初学者的网络数学游戏激增，但很少有程序或工具针对基本数学概念的理解增长，这对使患有LDM的年轻学生在数学和日常环境中熟练表现至关重要。Comps-A是一个计算机生成的教学程序，专注于解决加性应用程序问题；它将提供专门针对每个学生的学习档案实时定制的辅导。COMPS-A还将使推理和潜在的数学模型对他们更清晰，该工具的灵活性将促进在常规课堂环境中、在上学期间或课后的其他课程中或在家里进行小组或一对一教学。COMPS-A通过为学习困难的学生提供个性化和有效的RTI干预程序，解决了当今课堂上的一个重要实际问题。COMPS-A程序代表了一种基于数学模型的问题解决方法，强调理解和表达代数方程中的数学关系，从而支持广义问题解决技能的发展。COMPS-A将实现以下目标：1)创建课程内容、屏幕设计和教师手册，用于所有四个模块的加性文字问题解决；2)设计和开发跨平台的计算机应用程序，可以移植为基于Web的、iPad、Android或Windows应用程序，这种灵活性将使程序对所有学生都可用；3)进行小规模的单学科设计和随机对照试验研究，以评估COMPS-A提高学生应用程序解题成绩的潜力。以下研究问题将被解决：(1)COMPS-A计划与学生在加法数学问题解决中的表现之间有什么函数关系？(2)当学生对干预计划没有反应时，教师在识别学生的错误概念、替代推理和知识差距方面扮演什么角色？(3)必要的教学支架是什么，以解决学生的知识差距，从而促进学生的概念图式与问题解决所需的数学模型之间的联系，以促进对加法推理的有意义的理解和构建？COMPS-A的功能原型将被开发出来，随后将与一小群患有LDM的学生进行单学科设计研究，以实地测试最初的程序。最后，采用前测-后测对照小组设计，将参与者随机分配到小组，然后检验两种干预条件的效果：COMPS-A和一切照常。广泛的传播计划将使项目团队能够将成果分享给更广泛的社区，负责教育所有学生，特别是患有学习障碍的学生。

The Discovery Research K-12 program (DRK-12) seeks to significantly enhance the learning and teaching of science, technology, engineering and mathematics (STEM) by preK-12 students and teachers, through research and development of innovative resources, models and tools. Projects in the DRK-12 program build on fundamental research in STEM education and prior research and development efforts that provide theoretical and empirical justification for proposed projects.The 3-year exploratory project, Conceptual Model-based Problem Solving: A Response to Intervention Program for Students with Learning Difficulties in Mathematics, will develop a cross-platform mathematics tutoring program that addresses the problem-solving skill difficulties of second- and third-grade students with learning disabilities in mathematics (LDM). While mathematics problem-solving skills are critical in all areas of daily life, many students with LDM do not acquire key math concepts such as additive and multiplicative reasoning in a proficient manner during the early school years. In fact, about 5-10% of school-age children are identified as having mathematical disabilities which might cause them to experience considerable difficulties in the upper grades and experience persistent academic, life, and work challenges. Despite the proliferation of web-based mathematical games for early learners, there are very few programs or tools that target growth in the conceptual understanding of fundamental mathematical ideas, which is essential in enabling young students with LDM to perform proficiently in mathematical and everyday contexts. COMPS-A is a computer-generated instructional program focusing on additive word problem solving; it will provide tutoring specifically tailored to each individual student's learning profile in real time. COMPS-A will also make the reasoning and underlying mathematical model more explicit to them, and the tool's flexibility will facilitate group or one-on-one instruction in regular classroom settings, in other sessions during or after the school day, and at home. COMPS-A addresses a significant practical issue in today's classrooms by providing individualized and effective RtI intervention programs for students with LDM.COMPS-A program represents a mathematical model-based problem-solving approach that emphasizes understanding and representation of mathematical relations in algebraic equations and, thus, will support growth in generalized problem-solving skills.COMPS-A will achieve the following objectives: 1) Create the curriculum content, screen design, and a teacher's manual for all four modules in the area of additive word problem solving; 2) Design and develop the cross-platform computer application that can be ported as a web-based, iPad, Android, or Windows app, and this flexibility will make the program accessible to all students; and 3) Conduct small-scale single subject design and randomized controlled trial studies to evaluate the potential of COMPS-A to enhance students' word problem-solving performance. The following research questions will be resolved: (1) What is the functional relationship between the COMPS-A program and students' performance in additive mathematics problem solving? (2) What is the teacher's role in identifying students' misconceptions, alternative reasoning, and knowledge gaps when students are not responsive to the intervention program? (3) What are the necessary instructional scaffolds that will address students' knowledge gaps and therefore facilitate the connection between students' conceptual schemes and the mathematical models necessary for problem solving in order to promote meaningful understanding and construction of additive reasoning? A functional prototype of the COMPS-A will be developed followed by a single-subject design study with a small group of students with LDM to field-test the initial program. Finally, a pretest-posttest, comparison group design with random assignment of participants to groups will then be used to examine the effects of the two intervention conditions: COMPS-A and business as usual. An extensive dissemination plan will enable the project team to share results to a wider community that is responsible for educating all students and, especially, students with LDM.