In the Classroom

Inquiry-Based Learning

The need for more active involvement of mathematics students in the learning process is well documented. Involvement occurs through the use of inquiry-based learning. The curriculum materials that make up Discovering the Art of Mathematics (DAoM) reverse the typical lecture dynamic by being built on guided-discovery investigations.

DAoM materials focus on investigations, tasks, experiments, constructions, data collection and discussion prompts rather than transcribed lectures and worked-out sample problems followed by banks of routine exercises. The transformative impact this has on students in MLA classrooms can be seen clearly in the classroom vignette, student quotes, and videos shown below.

Inquiry-Based Classroom

What does inquiry-based learning mean to us? How do we teach? Let's walk into Volker Ecke's classroom and listen to his thoughts.

As the title of this blog post suggests I believe that inquiry helps tremendously in reaching learners on different levels at the same time - even if they all work on the same investigations. Why? First of all, students can work at their own pace.

multivariable calculus group discussing integration

In my perfect world students would be self-motivated, want to learn, collaborate with other students, ask lots of questions, pursue mathematics outside of class requirements, etc. What then is needed to make this happen? I believe this independent learning I am looking for relies on student curiosity.

Using name tags to group students

In the first class I ask students to write their name on a notecard and to place it in front of them. At the end of class I collect the name tags and in the beginning of the next class, I will set them up again, likely in different places. I call this strategy one of the “control knobs” of an IBL instructor, because where and with whom the students work has a big influence on the effectiveness of their learning.

Student Quotes

Perhaps the best way to understand the depth and powerful impact of our project is to read what the students have to say about their experiences. The following student quotes, collected during the project, are typical responses received as part of student journals, essays and reflections. Many more student quotes can be found in our quote library »

  • I feel if math is taught like this in elementary schools at the national level the sky is the limit for us.

  • I've done more math in this class than all of high school.

  • This class taught me how to think independently about not only math but other subjects and everyday problem solving.

  • The class was fair, and you really had to use your brain.

  • I never would have discovered .999… and 1 are equal to one other, I would have never made such beautiful string art and learned the mathematical principles considered with it, I would have never been able to study a famous mathematician, and learned so many essential things about life in general, without this class. The ability to do conjectures and proofs, to discuss things in a more expanded light, to be able to write out how I got something and truly know why, are qualities I didn’t have before this class.

  • When I figured out how to cut my own shape out I was actually excited. I actually had fun in doing it and was really proud of myself for being able to do it.

  • I will sort of miss this class -- most of the chapters of the book are interesting -- I'll keep the book for the fun of it.

Classroom Videos

The focus on active student involvement, student responsibility for learning, and decentralized role of the teacher is fundamental to this project. Without models or experience, such fundamental restructuring of one's mathematics classrooms can feel uncomfortable, foreign and daunting. Video clips from DAoM classrooms provide access into this culture of student discovery and inquiry.

Pick's Theorem

Caption: A small group interaction with their professor as part of investigations of the areas of polygons on a geoboard in relation to the number of pins contained within the polygon and those falling along the boundary. Mathematically, these are first steps towards the (re-)discovery and proof of Pick's Theorem. The students wondered which pegs actually lie on the boundary formed by the rubber band. Pedagogically, notice how the teacher works hard to support the students, drawing their attention to particular aspects of the geoboard. It seems he has a certain path in mind, given where the students are. Not all the hints are connecting with the students. In their attempts to describe the different patterns they see, the students realize that they can be grouped depending on the slope of the side of the triangle.

Powers of i

Caption: This video clip is from a section of  Mathematical Explorations, our Mathematics for Liberal Arts course. In this clip the students are exploring the pattern formed by powers of i and one of them turns to Prof. Hotchkiss as the authority for correct answers. As you watch this clip, listen to her questions and how Prof. Hotchkiss responds to them using inquiry-based questioning techniques.