The Inquiry-Oriented (IO) Approach and Advanced Mathematical Thinking (AMT) processes play an important role in improving the undergraduate math education. IO approach and AMT processes act as a new movement of modern math education based on the methods used in math high school education and lower-division courses of Calculus. Several published research papers talked about the Inquiry-Oriented Approach as a process that is pretty similar to any process of the Advanced Mathematical Thinking which combines both teacher activity and student activity, and they interact with each other to form what is known as inquiry process. According to the An Inquiry-Oriented Approach to Undergraduate Mathematics research paper in the Journal of Mathematical Behavior, Kwon and Rasmussen (2007) talk about the Inquiry-Oriented Differential Equations project as a collaborative effort to improve the undergraduate math education, and to study of how undergraduate math can draw on the theoretical and instructional advances initiated at the K-12 level, as well as, to create and sustain learning environment for powerful student learning. While there are several reasons for the initiation of the Inquiry-Oriented Differential Equations project such as the huge number of math departments at universities and colleges, the increase of student body diversity, and the decline in the number of math majors as mentioned by Kwon and Rasmussen, universities and community colleges need to implement this project as a fundamental way of improving the undergraduate math curriculum in general and differential equations curriculum in particular because this project can encourage the useful interaction between teacher and student through class activities and discussions in order to enable the teacher to measure the student mathematical thinking and how the teacher can form a new way of thinking based on what the student thinks.
In addition, Kwon and Rasmussen (2007) discuss several various characterizations of inquiry process in different research communities, for example, in general, inquiry can be identified as a set of assumptions using critical and logical mathematical thinking and considering alternative mathematical explanations, while in the philosophy of mathematics education, inquiry can be defined as the ability to learn how to "speak and act mathematically, participate in mathematical discussions, pose conjectures, and solve new or unfamiliar math problems" as mentioned by Kwon and Rasmussen . I totally agree with all those different characterizations of inquiry process because they are located under the title of how teacher inquiry into student mathematical thinking.
For example, when I taught Calculus II in Fall 2015, I implemented this inquiry process which was basically taken from my math book: A Friendly Introduction to Differential Equations because in Calculus II class, there are some topics that are taken from differential equations course such as finding the general solution of differential equations using separable method. Therefore, when we had an abstract topic in Calculus II such as partial fraction decomposition which is very commonly used in differential equations class, I told my students to think about a method that can save their time and can solve fifty percent of partial fraction decomposition problems. Some students told me that there are no other methods to avoid solving systems of linear equations to find the required constants, while others started giving me assumptions and suggestions about possible methods of doing partial fractions decomposition.
In that situation, as an instructor, I taught them how to think mathematically about other methods built on their assumptions and suggestions, and I kept telling them that there will be always a method to solve any mathematical problem if we learn how to creatively think about that mathematical problem based on what we learned from the traditional methods.
In conclusion, there are several advantages of applying the inquiry approach in our math classes such as differential equations, and its positive effect on the improvement of teaching and learning is one of the fundamental goals that we are looking for in our math classes because we want to provide our students with the tools for successful advanced mathematical thinking processes and the methods of reinventing mathematical ideas.
Kaabar, M. (2015, January 5). A Friendly Introduction to Differential Equations. Printed by CreateSpace, San Bernardino, CA, http://www.mohammed-kaabar.net/#!differential-equations-book/cuvt. Accessed on August 29, 2016.
Kwon, O., & Rasmussen, C. (2007). An Inquiry-Oriented Approach to Undergraduate Mathematics. Journal of Mathematical Behavior, 26(1), 189-194.