Generalization and abstraction both play an important role in the minds of mathematics students as they study higher-level concepts. In the second chapter of the Springer book Advanced Mathematical Thinking, 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 look out for opportunities to introduce generalization and abstraction in order to help our students better understand the pattern behind what they are learning.
Dreyfus says that numerous mathematical objects such as equations, numbers, and functions can be expressed in the classroom in the context of generalization in order to make students more comfortable with upcoming math topics. But it can take more mental effort for students to generalize concepts, and according 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 not born as mathematicians, but they are born with a brain that can be creatively enhanced by continuing the practice of generalization that can then lead to abstraction.
For example, when I taught Calculus II in Fall 2015, in the Telescoping and Geometric Series course lesson I taught my students how to use generalization by starting with a simple example of finding the first partial sums for 1+2+3+4+5+.., and then I talked about the relationship between partial sums and infinite series. This method introduces students to the mathematical concept starting from something simple and easy and then moving toward the more general underlying foundations.
Similarly, in the example of the washer method I described in my previous post here on the AMS Grad Student Blog, I can start with 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 begin to form their own generalizations by teaching them how to reconstruct this particular concept in a way that is easy to understand.
There are several advantages to applying generalization in our math classes, and its positive effect on teaching and learning is a fundamental way to provide our students with the tools needed for successful advanced thinking in mathematics.
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!