*This semester, I am co-teaching a graduate class focused on college learning and teaching. Each week, our students will have to write a reflection based on a prompt we provide them. Partially to “pre-test” each prompt but mostly because I believe in reflection and purposeful metacognition, I’ll write my own replies to the prompts in my blog.*

Aside from a minor concern about accountability (Did they read the assigned reading? Are we appropriately incorporating it?), the prompt for the first week of class is intended to do two things. First, it will start students thinking about one of the major topics of the course, bottlenecks in learning. This topic is being incorporated in this class by way of research conducted here at Indiana University (e.g. the History Learning Project) where faculty have identified ideas or concepts that are both essential and particularly problematic for students. Second, it will help ground our discussions of learning theory in students’ own experiences, both helping them understand the theories better and reminding them that these theories are not applied to just some students (an “other”) but to everyone, including themselves.

This week’s prompt:

This week’s reading focuses on the experiences and knowledge students bring to class and how that affects their learning. You, too, have experiences and knowledge you bring to this class that have shaped your beliefs about effective college learning and teaching. We will have to work to discover and incorporate those experiences, knowledge, and beliefs and that work begins with this reflection.

Reflecting on your undergraduate experience, which problem was more prominent in your major classes: Inaccurate prior knowledge, accurate but insufficient prior knowledge, or inappropriate prior knowledge? Why was it the biggest problem? Of the possible approaches described in the text, which could have been most effective in addressing the problem? Finally, do undergraduates in your discipline still have the same obstacles and would the same approach(es) work for them? Why or why not?

(I fear that my response may lack depth; my undergraduate degree is in mathematics and although I conduct quantitative research I have strayed very far from my undergraduate discipline and have become almost entirely a consumer of mathematical knowledge. But here goes…)

In my opinion, the biggest problem facing undergraduates in nearly any math class is that their prior exposure to math has given them a completely false image of math. For nearly everyone, mathematics is a large set of disconnected rigid rules and procedures that make little sense and are retained purely through repeated practice and memorization. In fact, the heart of math is creativity and connection. Math is the language of the universe but we never learn to read or speak it; instead, we focus on following often-meaningless rules and memorizing procedures without any context or explanation. Without understanding the role of creativity in mathematics and its inherent interconnectedness, no one can truly understand math and apply it well.

Of the methods presented in the text to help students “correct inaccurate knowledge,” the most applicable seems to be “ask[ing] students to justify their reasoning.” I believe that most students would not be able to justify their mathematical reasoning beyond “that’s just the way it is” which is not a very good reason. So the challenge – and it’s a big one! – would be to help students understand not merely *what* to do but more importantly *why* to do it. And that would include explaining that some mathematical conventions are just that – conventions that ensure consistency and help everything else make sense.

Finally, it seems obvious that this problem is not unique to me or my classmates but is a problem that has lasted for generations. Not only is it an inherently difficult and challenging problem but its history has given it momentum that is difficult to alter: Students who were never exposed to the inherent beauty of mathematics and creativity of mathematicians become teachers who never expose their students to those critical elements. And the challenge is passed on to the next generation.