Effective thinking

Effective Thinking Through Mathematics，这大抵是这二十多年来我听过的最好的课程了，为期九周，今天终于完成。

课程中举出了高效思考的五大原则：

• Understand Deeply，理解问题

  Don’t face complex issues head-on; first understand simple ideas deeply. Clear the clutter and expose what is really important. Be brutally honest about what you know and don’t know. Then see what’s missing, identify the gaps, and fill them in. Let go of bias, prejudice, and preconceived notion. There are degrees to understanding (it’s not just a yes-or-no proposition) and you can always heighten yours. Rock-solid understanding is the foundation for success.

  从简单情况开始，再推广到一般情形。找出问题的关键点，明确已知和未知，再找到中间的差异，再尝试填补差异。要摈弃偏差、偏见、先入为主的意见。理解是分程度的，任何时候都可以加深理解，坚实的理解是成功的基石。

• Make Mistakes，试错

  Fail to succeed. Intentionally get it wrong to inevitably get it even more right. Mistakes are great teachers - they highlight unforeseen opportunities and holes in your understanding. They also show you which way to turn next, and they ignite your imagination.

  敢于尝试，即便已知是错的，搞清楚错在哪里，为什么错，然后尝试纠正错误，这样就离正确更近了一步。

• Raise Questions，提问

  Constantly create questions to clarify and extend your understanding. What’s the real question? Working on the wrong questions can waste a lifetime. Ideas are in the air - the right questions will bring them out and help you see connections that otherwise would have been invisible.

  不停地提出问题，以理清、加深自己的理解。搞清楚当前问题的本质是什么，弄清楚问题经常能引出线索。

• Follow the Flow of Ideas，点子流

  Look back to see where ideas came from and then look ahead to discover where those ideas may lead. A new idea is a beginning, not an end. Ideas are rare - milk them. Following the consequences of small ideas can result in big payoffs.

  回顾已有进展，自己的想法/分支从何而来，可能走向何方。点子很稀有，要善于捕捉/酝酿。循着一系列小的点子，可能拼成大的突破。

• The Quintessential Element，跳出定势

  The unchanging element is change - by mastering the first four elements, you can change the way you think and learn. You can always improve, grow, and extract more out of your education, yourself, and the way you live your life. Change is the universal constant that allows you to get the most out of living and learning.

  唯一不变的就是变化。熟练掌握上述 4 点之后，还要持续总结、改进、提高。突破定势，走向成功！

课程中通过在数学方面的多个例子，演示了如何利用这五大原则去解决“不可能的难题”，启发非常大。后面我希望能在编程领域通过 leetcode 题库等方式，训练自己的思维方式，以期提升自己解决未知难题的能力。

创业又何尝不是这样？！