{"id":31116,"date":"2016-08-21T12:01:40","date_gmt":"2016-08-21T17:01:40","guid":{"rendered":"http:\/\/blogs.ams.org\/mathgradblog\/?p=31116"},"modified":"2016-08-21T12:01:40","modified_gmt":"2016-08-21T17:01:40","slug":"role-generalization-advanced-mathematical-thinking","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/mathgradblog\/2016\/08\/21\/role-generalization-advanced-mathematical-thinking\/","title":{"rendered":"The Role of Generalization in Advanced Mathematical Thinking"},"content":{"rendered":"

Generalization and abstraction both play an important role in the minds of mathematics students as they study\u00a0higher-level concepts.\u00a0In\u00a0the second chapter of the Springer book\u00a0Advanced Mathematical Thinking<\/em><\/a>, Tommy Dreyfus defines generalization as the derivation or induction from something particular to something general by looking at the common things and expanding their domains of validity. As we teach our own math courses, we can\u00a0look out for opportunities to introduce generalization and abstraction in order to help our students better understand the pattern behind what they are learning.<\/p>\n

Dreyfus says\u00a0that numerous\u00a0mathematical objects such as equations, numbers, and functions can be expressed in the classroom in the context of generalization in order to make students\u00a0more comfortable with upcoming math topics. But it\u00a0can take more mental effort for students to generalize concepts, and\u00a0according to my teaching experience, students tend not to try their best to generalize a mathematical concept if they do not receive good guidance from their teacher. I believe that students are\u00a0not born as mathematicians, but they are\u00a0born with a brain that can be creatively enhanced by continuing the practice of generalization that can then lead to abstraction.<\/p>\n

For example, when I taught Calculus II<\/em> in Fall\u00a02015, in the\u00a0Telescoping and Geometric Series<\/a>\u00a0<\/em>course lesson\u00a0<\/i>I taught my students\u00a0how to use generalization\u00a0by starting with a simple example of finding the\u00a0first partial sums for 1+2+3+4+5+.., and then I talked about the relationship between partial sums and infinite series. This method\u00a0introduces students to the mathematical concept starting from\u00a0something simple and easy and then moving toward\u00a0the more general underlying foundations.<\/p>\n

Similarly, in\u00a0the example of the washer method I described in my\u00a0previous post<\/a> here on the AMS Grad Student Blog, I can start\u00a0with a review about the volumes of disks, washers, and shells, and at the end use a real-life example to make it easy for them to find the volume of the given region. In this way, we can help students\u00a0begin to form their own generalizations by teaching them how to reconstruct this particular concept in a way that is\u00a0easy to understand.<\/p>\n

There are several advantages to\u00a0applying generalization in\u00a0our math classes, and its positive effect on teaching and learning is a fundamental way to\u00a0provide our students with the tools needed for successful advanced thinking in mathematics.<\/p>\n

Do you have any examples of how you have helped your students better understand a tough abstract or general mathematical concept? Share your experience in the comments below!<\/p>\n

 <\/p>\n

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Generalization and abstraction both play an important role in the minds of mathematics students as they study\u00a0higher-level concepts.\u00a0In\u00a0the second chapter of the Springer book\u00a0Advanced Mathematical Thinking, Tommy Dreyfus defines generalization as the derivation or induction from something particular to something … Continue reading →<\/span><\/a><\/p>\n

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