寻求计算与概念之间的一致性的数学意义建构：物理学导论的教学与评价
Mathematical Sensemaking as Seeking Coherence between Calculations and Concepts: Instruction and Assessments for Introductory Physics
摘要：
什么样的问题解决教学可以帮助学生运用所学知识解决未来遇到的新问题和不熟悉的问题？我们认为，数学意义建构，寻求形式数学和概念理解之间的一致性的实践，是成功的物理问题解决教学的一个关键目标。然而，典型的评估往往以更不相交的方式来衡量理解。为了捕捉学生解决问题的连贯性寻求的做法，我们引入了一个评估框架，突出了更灵活地使用这些解决问题的方法的机会。体现这种计算概念交叉框架的三个评估项目说明了连贯性如何推动灵活的解决问题的方法，这些方法可能更有效，更有见地，更准确。这三个评估项目被用来评估的教学方法的有效性，重点是发展认知意义理解技能。在一项准实验研究中，第一学期的三个平行讲座部分，介绍物理进行了比较：两个数学意义建构部分，一个有经验的教师（MS）和一个新手教师（MS-nov），和传统的教学部分作为对照组（CTRL）。在三个交叉评估项目上，数学意义建构学生比对照组学生更多地使用计算-概念交叉的方法，并产生更多的正确答案。在课程结束时，学生对解决问题的连贯性的认识论观点预测了他们的交叉方法的使用，但没有充分考虑MS和CTRL组之间交叉方法使用的差异。这些结果说明了新的教学和评估框架的研究数学意义和适应性问题解决的专业知识。
英文摘要：
What kind of problem-solving instruction can help students apply what they have learned to solve the new and unfamiliar problems they will encounter in the future? We propose that mathematical sensemaking, the practice of seeking coherence between formal mathematics and conceptual understanding, is a key target of successful physics problem-solving instruction. However, typical assessments tend to measure understanding in more disjoint ways. To capture coherence-seeking practices in student problem solving, we introduce an assessment framework that highlights opportunities to use these problem-solving approaches more flexibly. Three assessment items embodying this calculation-concept crossover framework illustrate how coherence can drive flexible problem-solving approaches that may be more efficient, insightful, and accurate. These three assessment items were used to evaluate the efficacy of an instructional approach focused on developing mathematical-sensemaking skills. In a quasi-experimental study, three parallel lecture sections of first-semester, introductory physics were compared: two mathematical sensemaking sections, with one having an experienced instructor (MS) and one a novice instructor (MS-nov), and a traditionally-taught section acted as a control group (CTRL). On the three crossover assessment items, mathematical sensemaking students used calculation-concept crossover approaches more and generated more correct solutions than CTRL students. Student surveyed epistemological views toward problem-solving coherence at the end of the course predicted their crossover approach use but did not fully account for the differences in crossover approach use between the MS and CTRL groups. These results illustrate new instructional and assessment frameworks for research on mathematical sensemaking and adaptive problem-solving expertise.
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