*By Sarah E. Andrews and Justin R. Crum, undergraduate Mathematics majors at Northern Arizona University, and Taryn M. Laird, graduate student in Mathematics at, and 2014 graduate of, Northern Arizona University.*

*Editor’s note: The editorial board believes that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. We thank Ms. Andrews, Mr. Crum, and Ms. Laird for contributing their essay. More information regarding inquiry based learning can be found at **http://www.inquirybasedlearning.org/**.*

Inquiry based learning (IBL) classes inspired each of us to believe that we could go into mathematics. That we belonged. We may be able to prove something important or make an impact in the lives of other budding mathematicians. IBL classes have given us this confidence to believe in ourselves, and to have fun trying to discover for ourselves what math is and where it will lead us. It was not only this sense of being able to discover, however, it was also learning how to collaborate with others. Mathematics is not an isolated endeavor, but rather a concentrated attempt by groups of people working toward their common goal. In normal lecture-based classes, we would talk to our friends, and if we got stuck, we might ask one another what to do next. In the IBL classes, we would talk to each person in the class. Students would ask each other questions willingly. We would make new friends, and ask more questions, until each of us decided we were satisfied — we understood the material now.

In a typical math classroom, we go in, listen to a lecture about the infinite number of primes, or the procedure for finding a subgroup, and frantically copy the deluge of mathematics, all while trying to comprehend the complex concepts being presented. Then we go home, eat dinner, and attempt to apply our concepts to similar, but often new, ideas, without the guiding hand of a trained mathematician. In IBL classes, we receive materials, go home, and actively think about the new definitions and theorems. How do we apply these, how can we prove those, where can these be used to prove other statements? Then we marinate in these concepts, think about the math behind them for the rest of the day, and once we show up in class again, we are ready to discuss. We are ready to ask questions to further our understanding of the concepts, and attempt formally proving theorems using these concepts. This format motivated us to do more, to actively participate in the mathematics happening around us, and to think critically about theorems, rather than accepting them at face value.

The critical thinking that IBL instilled in us led to a deeper understanding of the mathematical process. There was creativity required; failure does happen, but you get up and try again. It was no longer black and white, it was a science — one that had experiments you could fail, but after having learned a lesson from the failure, you persevere and find a solution. From this, we learned proving mathematical results is a science experiment, at least when students are allowed to become active participants. We wanted to beat the problems, especially if they were challenging. We would sit there and think about why is the center of a group a subgroup of the overall group? Then, instead of giving up, we put various ideas on paper. Wrote down assumptions and useful theorems. How can these be applied? We got stuck. Instead of giving up though, or getting frustrated, we kept going. We emailed professors, got hints, and pushed forward. Then come class day, we could present a valid proof.

Looking back, this happened all the time. We would be at the library, banging our heads against the white boards in a study room writing down assumptions and wondering where to go. Normally, this is where we would stop. Where we would get frustrated. Where most students would give up, wait to go to class, and ask the instructor how to finish. However, in each IBL class, this is not the point where we gave up. This is the point that students strove for, to prove that they could get past that portion of the hardest problem on the set.

Becoming active participants opened the door to the mathematical community. We talked to professors and created a sense of community within our IBL classes that led to a deeper involvement within our department. We forged necessary relationships with our professors, and they instilled a drive to know the why behind the math we discussed. Instead of sitting in class watching the professors present a theorem and blindly accepting it at face value, we would question it. Another student was at the board trying to convince us that induction was a valid method of proof. Instead of just using the method and deciding that, yes, it worked, we could decide for ourselves if we believed what was written. We could critically think about the math being put on the board instead of just assuming that because the person is a professor, there is no need for them to convince us. After IBL classes, each of us have gone on to do some sort of undergraduate research or independent study. Taryn Laird specifically went on and did research with one of her IBL instructors. We became experienced independent learners who were driven to learn more and ask questions, and the relationships that we had forged were key in this process.

IBL classes gave each of us a glimpse into the real world of mathematics. It transformed our thinking, and gave us the extra confidence that we could do math. With this being said, much like the academic world of mathematics, our classes were predominantly male. Before IBL classes, this feeling of gender bias ranged from bothering us from the back of our minds, to being shoved down our throats. The IBL classes gave us confidence that, male or female, we could do math. When we went through these classes the instructors fostered an environment where we could ask all the same questions, come up with all the same answers, and prove all the same theorems. There are no gender reasons that should be able to stop us — each of us are just as capable as the others. Despite the obvious discrepancy in amount of males to females, we realized that anyone that told us math is not a woman’s field was wrong.

No lecture can compare to an IBL class. Lectures are useful, but IBL classes can transform students and create positive environments for people learning to question the world around them. They take students in that enjoy math, and allow them to fall in love with it completely. It helps people become less shy, gain necessary working and life skills, and truly learn to critically think. Inquiry based learning classes are about participating, learning, and growing as a mathematician.

Great paper, Taryn and Justin!

Your enthusiasm comes across very clearly, giving the paper life — even excitement (do mathematicians know that word? :-))! It sounds like you received an excellent education in your IBL classes. I suspect this was largely, or at least partly, due to passionate and low-ego professors who were able to create an environment that encouraged the students to participate more fully and more creatively in the learning process. And that’s a very valuable treasure to have experienced!

TG