Promoting Active, Inductive, and Computational Learning in Mathematical Methods for Physicists

摘要许多物理和工程课程都包括一门数学技巧课程。该奖项帮助开发和测试这类课程的一套练习，以便学生更好地理解材料的动机，并帮助学生在课程中融入计算机技能。例如，励志练习引导学生完成在屏幕上制作图形动画(针对矩阵)或分析恒星位置数据以显示太阳系以外行星运行的证据(针对傅里叶级数)所需的步骤。在每一种情况下，学生都会被引导使用熟悉的数学推导，最终达到一个点，在那里他们可以看到对新的数学技术的迫切需要。计算机辅助练习引导学生完成一些任务，比如比较方程的精确解和近似解，或者解复杂的方程并绘制出解的曲线图。这些练习有助于阐明数学原理，同时在具有挑战性和趣味性的问题背景下提高学生的数字技能。这类课程具有独特的挑战。首先，学生们通常在学习一年或更长时间后才使用开发的数学工具。教育研究表明，在有意义的环境中学习的材料最容易被记住，当在相关的环境中再次出现时，材料被回忆起来的效果最好。因此，这个项目中正在开发的动机练习有望提高学生的记忆力。此外，在我们的数字时代，这些课程的内容应该经常更新，因为传统的技能(如手工求解方程组)往往不如知道如何有效地使用计算机来解决问题那么重要。这个项目正在开发的练习将在全国范围内产生广泛的影响，因为教授们将得到帮助，修改他们的课程，在教授传统数学技能的同时教授计算机技能。编写的材料正在网上免费提供，并在全国会议上展示。此外，不同机构的一些参与教职员工自愿在开发这些材料时使用它们。这些练习的有效性是通过在引入练习之前和之后的学期测试来评估的，计算技能在课前和课后进行测试，并利用调查来收集学生和教职员工的反馈。

AbstractMany physics and engineering curricula include a course on mathematical techniques. This award assists in the development and testing of a suite of exercises for this type of course so students better understand the motivation for the material and to help students incorporate computer skills within the course. For example, motivational exercises walk students through the steps needed to animate graphics on a screen (for matrices) or to analyze data on star positions to show evidence of orbiting planets beyond our solar system (for Fourier series). In each case the students are led through derivations using familiar mathematics to eventually reach a point where they can see the vital need for a new mathematical technique. The computer assisted exercises lead students in tasks such as comparing exact and approximate solutions to equations or solving complicated equations and plotting the solutions. These exercises help illustrate mathematical principles while simultaneously enhancing student numerical skills in the context of challenging and interesting problems.Courses of this type pose unique challenges. First, the students often don't use the math tools developed until a year or more after they learn them. Educational research shows that material is retained best when it is learned in a meaningful context and best recalled when it comes up again in a related context. The motivational exercises being developed in this project are therefore expected to improve student retention. In addition, in our digital generation the content of these courses should be frequently updated, since traditional skills (such as solving systems of equations by hand) are often not as important as knowing how to effectively use a computer to solve the problems. The exercises being developed in this project will have a broad national impact as professors are assisted in revising their curricula to teach computer skills alongside traditional math skills.The materials developed are being made freely available on the Web and presented at national conferences. Moreover, a number of participating faculty members at different institutions are volunteering to use these materials as they are developed. The effectiveness of the exercises are being assessed with tests given in semesters before and after the introduction of the exercises, computational skills are tested before and after the class, and surveys are being utilized to gather student and faculty feedback.