Encourage Deeper Inquiry Early in the Math Major

To help students develop deep understanding of mathematics, Dr. Kristin Camenga asks them to create their own questions-and embrace failure.

Traditional math instruction often focuses on product (arriving at the correct answer) rather than process (the logic applied to arrive at that answer). While this can be effective-after all, there is a reason that untold numbers of students throughout the ages have learned mathematical processes by performing them over and over again-it can also deprive students of developing an understanding of the deeper meaning of their work.

“This [type of instruction] puts us in a position where students are always receiving information,” says Kristin Camenga, PhD, associate professor of mathematics at Juniata College in Pennsylvania. “But we need students to see how the process actually works rather than simply having them solve contrived problems.”

To battle the product-focused approach, Camenga has become a proponent of assigning what she calls “problem-posing inquiry projects,” which allow her to steer instruction away from teacher-created, teacher-assigned problems. In this way, she empowers students to learn how to ask their own questions-and, as a result, to develop their curiosity and become independent as mathematicians.

Camenga’s problem-posing inquiry projects are a version of Inquiry-Based Learning (IBL)-an active-learning approach that begins with scaffolded mathematical tasks. Camenga then encourages students to formulate their own questions, research the potential answers, and share the results with the larger group.

For example, Camenga says, “students find new questions by looking at ways that the problem could be changed: What if the numbers are changed? … What if we go up to a higher dimension (3×3 instead of 2×2 matrices)? Can we generalize the problem so that the original problem is an example? Can we use the structure of our original solution to find other examples or shapes that use the same method? … Or is there something about the structure of the original problem that lets us pose a problem that can’t be solved by the original method, thus pushing us to try something new?”

In this way, mathematics concepts and mathematical problems become the raw material for student inquiry-so they represent the starting point, rather than an end point. “What I’m asking students to do is think about what they need to prove rather than simply giving them a theorem,” Camenga says.

Course description: Number Theory including divisibility, primes, congruence, congruence equations, quadratic residues and quadratic reciprocity, arithmetic and multiplicative functions, Diophantine equations, and other topics selected according to interest.

Camenga says her problem-posing inquiry approach works best in courses where students have computational work to do-that work becomes raw material for further conjecture. She uses the practice in courses such as Number Theory, Discrete Mathematics, Combinatorics, higher-level geometry courses, and in mathematics research seminars.

“For me, this approach just makes sense because math is very iterative,” she explains. That said, it is very different from traditional instruction, and it requires a different mindset on the part of instructors-as well as different preparation-in order to be effective.

Here, Camenga gives a few suggestions for any instructor looking to adopt the practice of problem-posing inquiry projects.

Become comfortable with losing control

“Professors want to keep a sense of control over how their students progress through a course because they want to ensure that they get through everything on the syllabus,” Camenga explains. She says that she does make sure students understand all the necessary theorems, but her approach offers more than that, too. “If [you] value that students are thinking through the work themselves-not just believing what the instructor tells them, but coming up with the results on their own-[you] have to become comfortable with being uncomfortable.

Because of this, I am more likely to use this approach in courses where the syllabus is less standard,” she adds. “I can decide on the bare-bones essentials I want students to learn and cover this in a more ‘standard’ IBL way, with scaffolded tasks, and then leave some of the other areas for more student investigation.”

Build relationships to ease student fears

For many students, taking an inquiry stance and posing their own questions (rather than sticking to teacher-created questions) feels risky. “If they feel safe with you, it’s entirely different,” Camenga says. “In math, the distance between the teacher and the student can be huge. I take every opportunity to connect with students and build that rapport that enables them to feel safe approaching math in this new way.”

For example, she recommends building rapport by setting up feedback loops, such as emailing with students and providing incentives for them to attend office hours. She also uses these techniques to track her students’ progression through course material.

Make the process part of grading

While students are working through initial problems, Camenga asks them to log the number of hours they spend, then makes that part of their grade. “Even though I want the product to be correct, I work to make the grading a lot about the process, because the effort is important,” she says.

After Camenga’s students solved some problems, she models for them the process of finding new problems to consider. To aid them in doing so, she asks students to consider their previous work, then asks what they have noticed about those problems and what they are curious about. This simple directive can help students find entry points into further inquiry.

She evaluates this by having them write her periodic emails about their mistakes and failures in the class. “I also do certain things such as dropping lowest grades to build students’ confidence with productive failure and help them feel OK with growth mindsets,” says Camenga.

Get departmental buy-in

Camenga suggests connecting with other faculty to socialize the idea that it is important for students to develop inquiry-based dispositions and growth mindsets in addition to problem-solving acumen. “It’s easier to implement [this approach] when your department believes students should actively take part in this process, and when other faculty are building in opportunities for students to engage in inquiry-based problem-solving in their courses as well,” she explains.

One compelling benefit of problem-posing inquiry projects that she has shared with colleagues: These activities have helped her get a better sense of her students’ thought processes, which makes her a better teacher. “It gives me information about what they’re doing and how they’re doing it, which informs my instruction,” she explains.

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