What Do Hobbits Know About Mathematics?

Sam: If I take one more step, I’ll be the farthest away from home I’ve ever been.

Frodo: Come on, Sam. Remember what Bilbo used to say: ‘It’s a dangerous business, Frodo, going out your door. You step onto the road, and if you don’t keep your feet, there’s no knowing where you might be swept off to.’

For many students, it is scary to be pushed to think differently about mathematics or to participate in a different type of classroom environment (for example, a flipped classroom, IBL classroom, active learning classroom, etc.).  These new experiences create a certain level of discomfort in adapting to new styles and expectations, which makes it easy to pine for the comfortable ways that math has “always been taught.”  Of course, this emotional response can be just as strong for teachers as it can be for students.

In the end, we want our students to gain a deeper understanding of mathematics.  It can be easy to think we need to take every student on a grand adventure like the Hobbits in The Lord of the Rings, to show them how to battle (mathematical) orcs or dragons, and to bring them to a crowning achievement of casting the one ring (perhaps with unity) into the fires of (mathematical) Mount Doom.  But maybe that isn’t what the students need, especially at the beginning of their college careers.  Maybe they just need us to encourage them to go one step further in their mathematical journey than what they had previously thought was possible.  In this post, I would like to highlight a few of my favorite articles that have centered on the theme of creating dynamic and supportive learning environments where students can get swept away in mathematical exploration and play.

Ben Braun wrote a brilliant article about using open problems as homework.   I reread this post at least once per year and continue to find inspiration in it.  If we want students to start thinking like mathematicians, and if we want to share the joy of mathematics with them, then why not show them problems whose solution cannot simply be found in the back of a textbook? Why not push them to think deeply about a problem on their own, rewarding them for the effort they have put forth rather than for getting the “right answer”?  It is unlikely that anyone will solve an unsolved problem, but it is likely that someone will become more excited about mathematics.

Related to this theme of discovery, Lara Pudwell wrote about an experimental math course she has developed, where students take ownership of problems that they explore and investigate on their own.  Students engage in a journey of mathematical discovery that is typically reserved for research experiences and get to see beautiful mathematics that does not always make its way into the undergraduate curriculum.

However, as Bilbo reminds us, we also need to teach students how to keep their feet beneath them through this new experience of learning mathematics. Jess Ellis Hagman shared important lessons on working with students from marginalized groups in an active setting, and Jessica Deshler shared practical tips about promoting gender equity in the classroom on the MAA Teaching Tidbits blog.  Art Duval’s post on kindness is one of the most beautiful pieces I’ve read recently, reminding us that teaching mathematics is as much a human endeavor as a scientific one.

To me, these posts are inspiring because they show how to incorporate mathematical adventure into the student experience, while also reminding us that the journey is difficult and the road is tough for many students.  Lessons of kindness and grace, coupled with an understanding of how to balance learning styles, personality types, and issues of identity within groups are important for creating a mathematical adventure that is engaging and inviting for all students.

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