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

探索研究K-12项目（DRK-12）旨在通过研究和开发创新资源、模型和工具，显著提高pre -12学生和教师对科学、技术、工程和数学（STEM）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依据。这个为期3年的探索性项目名为“基于概念模型的问题解决：对数学学习障碍学生干预计划的回应”，将开发一个跨平台的数学辅导计划，以解决数学学习障碍（LDM）的二年级和三年级学生解决问题的技能困难。虽然数学解决问题的能力在日常生活的各个领域都是至关重要的，但许多LDM学生在上学的早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%的学龄儿童被确定为有数学障碍，这可能会导致他们在高年级遇到相当大的困难，并经历持续的学业、生活和工作挑战。尽管针对早期学习者的基于网络的数学游戏越来越多，但很少有针对基本数学概念理解的程序或工具，这对于使患有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，侧重于加法解题；它将根据每个学生的学习情况实时提供量身定制的辅导。COMPS-A还将使推理和基础数学模型对他们更加明确，并且该工具的灵活性将有助于在常规课堂环境中，在上学期间或放学后的其他课程中以及在家中进行小组或一对一的指导。COMPS-A通过为LDM学生提供个性化和有效的RtI干预计划，解决了当今课堂上一个重要的实际问题。COMPS-A项目代表了一种基于数学模型的问题解决方法，强调对代数方程中数学关系的理解和表示，因此，将支持广义问题解决技能的增长。COMPS-A将实现以下目标：1)创建所有四个模块的课程内容、屏幕设计和教师手册；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， iPad， Android或Windows应用程序，这种灵活性将使所有学生都可以访问该程序；3)进行小规模单受试者设计和随机对照试验研究，评估COMPS-A提高学生单词问题解决能力的潜力。本文将解决以下研究问题：(1)COMPS-A项目与学生加法数学解题成绩之间的函数关系是什么？(2)当学生对干预计划没有反应时，教师在识别学生的误解、替代推理和知识空白方面的作用是什么？(3)为了促进对加法推理的有意义的理解和构建，有哪些必要的教学框架可以解决学生的知识空白，从而促进学生的概念方案与解决问题所需的数学模型之间的联系？COMPS-A的功能原型将开发出来，然后由一小群患有LDM的学生进行单主题设计研究，对初始程序进行现场测试。最后，采用前测后测的比较组设计，将参与者随机分配到不同的组，然后用于检查两种干预条件的效果：COMPS-A和一切照旧。广泛的传播计划将使项目团队能够与更广泛的社区分享成果，该社区负责教育所有学生，特别是患有LDM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.