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

许多物理学和工程学课程都包括数学技巧课程。该奖项有助于开发和测试这类课程的一套练习，以便学生更好地了解材料的动机，并帮助学生将计算机技能融入课程。例如，动机练习引导学生完成在屏幕上动画图形所需的步骤（对于矩阵）或分析星星位置的数据，以显示太阳系以外的行星轨道的证据（对于傅立叶级数）。在每一种情况下，学生们通过使用熟悉的数学推导，最终达到一个点，他们可以看到一个新的数学技术的重要需求。计算机辅助练习引导学生完成比较方程的精确解和近似解或求解复杂方程并绘制解等任务。这些练习有助于说明数学原理，同时提高学生在具有挑战性和有趣的问题中的数学技能。这类课程具有独特的挑战性。首先，学生们通常在学习了一年或更长时间后才使用开发的数学工具。教育研究表明，在有意义的背景下学习材料时，记忆效果最好，在相关背景下再次出现时，记忆效果最好。因此，该项目中正在开发的激励练习有望提高学生的保留率。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（如用手解方程组）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。该项目正在编制的练习将产生广泛的全国影响，因为教授们将得到协助，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，编制的材料将在网上免费提供，并在全国会议上介绍。此外，不同机构的一些参与教师自愿使用这些材料。这些练习的有效性正在评估与测试之前和之后的学期介绍练习，计算技能进行测试之前和之后的类，并正在利用调查收集学生和教师的反馈。

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.