在这篇文章中,我想分享一些观察about how my approach to learning has changed since I started working as a professional mathematician, and how I have changed my approach to teaching with the hope of helping my students develop more effective and relevant learning strategies earlier in their mathematical journeys.
Reading has become my primary mode of knowledge acquisition.
When I was an undergraduate student, I rarely read mathematics. It wasn’t from a lack of interest in the subject. I remember being enthusiastic about my courses and the joy I felt from solving problems. I simply didn’t read much mathematics. Not真无论如何。我读到的最接近的东西是通过教科书扫描一个主题或定理，可以帮助我链接两个将允许我解决家庭作业问题的概念。拼图阅读数学的拼图方法持续良好进入研究生院。它与我目前的情况有显着差异，我学习的大多数新数学，我从阅读中获得。那是什么改变了？必要性。作为教授，我可以参加研讨会和会议，以了解有关某些科目的更多信息，但不是与参加课程相当的程度。相反，我花了很多时间学习，并且可用的格式几乎总是写文字。
当我还是一个学生,读简单的需要’t there. I was fortunate to study at an undergraduate program with many dedicated teachers, who prepared clear, accessible lectures and class activities, so I could successfully complete my coursework without doing the assigned readings. It didn’t become an issue for me until I was a graduate student when I had to look up details of proofs that didn’t fit into lecture notes and read lots of articles for my dissertation research. It was a difficult transition for me.
当我还是学生时，我花了很多时间学习新技术，努力解决问题集......然后立即忘记它们。我不认为我特别没有动漫或懒惰 - 当他们说某些概念很重要时，我完全相信我的教授 - 然而，我忘记了在学习之后很久了解到的东西。发生了什么事？一方面，自然是在阶段学习新事物，在第一次遇到的内容下拿起一小部分，然后是每次随后曝光的新信息。在过程的每个阶段，我们内化了一部分并忘记了我们观察到的其余部分。
另一方面,我认为我的努力是合适的婚姻对象ally related to context. I remember my professors giving clear explanations for why different techniques were developed and how they were used in practice, but there was a disconnect for me because I didn’t have any personal experience developing mathematical techniques, nor did I have an application of my own in mind. Looking back over my career, my most productive learning experiences have come from working on a project where I didn’t have all of the tools I needed and had to learn them on the fly in order to complete the project. In those cases, I didn’t watch a tutorial or listen to a lecture about standard techniques and then practice them on a variety of examples; I started with the problem I was trying to solve, found a technique in the literature that was used to solve similar problems, and figured out how to apply or adapt the technique to my particular situation.
All of my best ideas have had humble beginnings.
When I was a student, I had a growth mindset about mathematical knowledge but a fixed mindset about mathematical creativity. I believed everyone could have positive, successful, and meaningful experiences with mathematics by learning new techniques but mathematical creativity was an inherent ability that could not be developed. I don’t know why I felt this way, and I can’t recall anyone ever telling me it was the case, yet I remember that impression weighing on me a lot. Whenever I worked on homework sets with other students and someone would figure out how to solve a problem I was stuck on, I always assumed it was because they had some amazing insight that I would not have been capable of finding on my own. I was so preoccupied with trying to figure out whether or not I had what it took to become a successful mathematician that it never occurred to me to ask them how they came up with their idea. Consequently, I spent a lot of time feeling frustrated, not being particularly productive, and waiting for inspiration to strike because that was where I thought creative solutions originated.
现在,我有更多的经验——confidence that comes with it – I can recognize that all my best ideas started with simple observations. And while there is no clear-cut recipe for creativity and innovation, there are concrete things I can do to cultivate situations that make those important kernels of ideas of possible. Instead of dwelling on what I don’t know how to do, I focus on exploring what I can do that might produce a new insight, such as writing out some examples, constructing a conceptual diagram, or drawing a picture.
All of my proudest accomplishments were made possible through the generous help of people whose experiences and perspectives are different from my own.
作为一名学生，我发现在团队中的工作很少，尤其是陌生的队友。Team assignments typically went one of two ways for me: either I was confident in my abilities and did the vast majority of the work or I was insecure about my abilities, didn’t want to look stupid, and held back my ideas thinking it was better to appear ignorant than open my mouth and confirm it. In the former scenario, I didn’t mind doing most of the work because I was confident in my ability to succeed, and it often seemed easier to do most of the work myself rather than try to coordinate my teammates’ efforts.
When I look at how the accomplishments I’m most proud of have come about, and how much I have learned in recent years working at an international college in Singapore, I can’t help but wonder how many opportunities to learn and grow I missed out on because I simply wasn’t looking or I didn’t appreciate how much effort goes in to building enough trust to open up a beneficial exchange of ideas.
Changing the Way I Teach
Here are a few ways I have changed my approach to teaching in response to these observations.
对于初学者来说，我不再依赖讲座或视频教程来呈现新想法。相反，狮子的内容交付份额以阅读作业的形式出现。为了支持我的学生，因为他们适应这个模型，我使用社会注释平台，Perusall, which allows them to highlight passages and ask questions, contribute or link alternative explanations, and propose solutions to “check your understanding” type exercises. They can also upvote annotations of their peers that they find helpful. In addition to developing technical reading skills, Perusall offers the valuable practice contributing to social media debates and online forums likeStackexchange.in a safe and controlled environment.
To offer my students an authentic learning environment that emulates the typical “on the job” learning that takes place in many technical professions, I have started to build each of my courses around three or four substantial team projects. Instead of asking students to master content and then apply what they have learned to a bigger project, I design the projects in a way that prompts students to learn the relevant material as they go. Each project is assigned on the first day of its respective segment of the course. The students are typically able to understand what the project prompts are asking but are not aware of any obviously relevant tools to get started.
为了促进有效的团队合作，我已经采用了Team-Based Learning(TBL) model, where each lesson has a reading assignment to be completed before class, individual and team readiness assurance tests at the start of each class, and a substantial problem-solving session that enables students to apply and extend their understanding of the tools they will need to successfully complete the project. Students take the readiness assurance tests and work together on the problem-solving sessions within their project teams throughout the duration of the project in order to develop a productive group dynamic.
为了鼓励和加强数学研究和创造力的良好习惯，当他们展示生产性研究过程的重要因素时，我开始承认和赋予团队的信誉，例如生成示例，识别模式，提出问题和制作猜想，以新的测试猜测信息，文献中相关主题之间的信息，以更好地了解手头的问题，并根据初步发现重新评估一种方法。由于许多这些元素可能难以辨别在最终写的报告中，因为我已经开始询问每个团队在整个项目中提交文件的活动日志。我的活动日志的标题受到严重影响关于证明的创造力 - 正在进行的量规。
Finally, in addition to research and creativity, I have started to encourage and reinforce good habits for effective and respectful team interactions by asking each team to prepare a mission statement during the first week of the project where they agree on a team name, tentative work schedule, and initial plan of attack. I also ask each team to prepare a set of guidelines for how they will conduct their meetings and a set of criteria for how they will evaluate each other’s contributions to the project.
The idea for creating guidelines came from my experience facilitatingIntergroup Dialogue(IGD) at Yale-NUS College. IGD is a structured conversation between members of different social identity groups that encourages participants to explore singular and intersecting aspects of their identities while critically examining dynamics of power, privilege, diversity and inequity in society. Because the dialogues can be difficult or contentious, a lot of the groundwork for IGD aims at building trust and creating a space in which people can share their ideas freely without judgment. For instance, at the beginning of each dialogue, the participants prepare a list of guidelines. I adapted those guidelines to fit a team-based learning classroom: The IGD guideline“We all recognize that participation in this dialogue is voluntary. Everyone who is here wants to be here.”变成了“我们都认识到这门课程是一项选修课。这里的每个人都想在这里。“Most of the guidelines are common sense statements, but articulating them in a mission statement provides avenues for students to speak their discomfort and overcome obstacles in a responsible and respectful manner.
Sample Project Brief
Here is the firstproject brieffrom my Discrete Mathematics course, which is typically taken by second-year prospective MCS majors at Yale-NUS College whose primary interest is computer science. The course meets twice per week for 110 minutes at a time. Each lesson consists of a pre-class reading assignment (8-10 pages of text, approximately 2 hours of interactive reading), in-class readiness assurance tests (20 minutes), and an in-class problem-solving session (90 minutes). The project spans six class meetings, including one lesson each on the Pigeonhole Principle, mathematical induction, and basic enumeration, two lessons on combinatorial proofs and bijections with emphases on the Binomial Theorem and Fibonacci numbers, and one class meeting designated as work time so students have a full week free of reading assignments and problem sets to complete their reports.
[The project] was actually a very fun and enjoyable experience, while also providing a good amount of challenge and difficulty. When we first received the project brief, we were genuinely stunned by what we had to do – we didn’t really know where to begin, and everything we tried seemed to be useless. But it was really nice to see us slowly progress, picking at the problem bit by bit, sometimes with no results, sometimes with huge chunks of the problem falling off. I really saw the advantage of having very different minds work on the same problem. I believe my teammates and I, having come from different backgrounds in terms of interests and experiences, approached the problems quite differently, and we were able to really complement each other and bounce off each other’s ideas. All of us contributed in big ways, and together we managed to come out with a closed formula pretty early into the project. Eventually, we managed to link the closed formula to one of the combinatorial objects, and quickly pieced bijections together. Even in the final moments of the project, the group shone through as we all picked on different parts of the project, trying to polish it off as well as we could.
Other students struggled at times, but eventually warmed up to the approach. For instance, a student wrote:
Initially, I felt rather excited about tackling the questions. We made some observations that turned out to be insightful and it felt like the project was going in the right direction. When the team and I got stuck at the later stages of this project, I became frustrated and lost motivation. But my teammates continued to encourage me and kept trying to develop new methods of solving the problems. Through this project, I learned that solving problems is not always a smooth path. It is helpful to acknowledge our frustration and to expect difficulties so that we are less anxious when we are stuck.
在整个项目中，我已经了解了很多关于数学推理如何发生的事情，并且[i]对我对数学是如何完成的看法发生了很多。被用来在高中中的常规问题解决方法，只有一个正确的答案和一些最适合确定这个答案的预设方法，我已经开始爱这个项目中采取的合作方法和整体一般离散数学课程。这是一个充满活力的学习环境，我非常感谢拥有我的团队成员的支持和知识。我一直觉得我的数学推理技能不如其他高级学生或人的方式更快地推理，更清晰，更优雅，但我已经开始了解最终抛光的产品并非所有这些都似乎 - 这是它的过程是最重要的，在我正在努力提高我可以用来改进最终解决方案的技能时，我可以在过程中贡献很多东西。
Despite these successes, there is still a lot of room for improvement. A common piece of critical feedback I receive from students is that the reading assignments are very difficult, even with the added support from Perusall. There are a lot of factors at play here such as the choice of text, size of the class, and my (in)ability to effectively respond to the Perusall discussions in real time.