I’m up for contract review this year, and I’m also eligible for promotion, so I just completed my teaching narrative. I got **real** self-indulgent on this thing and ended up writing almost 3500 words (yikes; sorry, committee). There’s one section in particular that I thought would be useful to share.

In July I participated in an AIBL workshop and just enjoyed the hell out of it. Even though I’ve been an active learning devotee for most of my career, and thus have been practicing “big-tent IBL” for a number of years, I learned a ton and developed my confidence to go with a full-on, notes-only, student-presentation style analysis course this semester. I really can’t recommend AIBL workshops enough.

One thing that particularly resonated with me was how the workshop facilitators framed IBL in terms of the presence and interaction of **four pillars**. As soon as I heard them, I knew that they were going to be a really useful way to explain my teaching philosophy; as soon as I started writing, a lot of my attitudes and heuristics fell neatly into the framework. Maybe this will spark some similar thoughts for you in your own teaching practice and philosophy.

**Students engage deeply with coherent and meaningful mathematical tasks.**- There’s a lot to unpack here. First, in a math class, we need to be learning math, so our tasks must be
**mathematical**. What tasks “count” as mathematical? That is, what tasks honestly reflect the actual practice of working mathematicians? What tasks help students develop honestly mathematical habits of mind? My answers to these questions are always evolving, but in general, I try to focus on understanding over memorization, and on concepts rather than procedures. If there is a computation, I ask students what it**means**when they’re done. - Tasks must be rich enough to support
**deep engagement**. This is another reason why I deemphasize tasks whose sole purpose is computation: a student can turn off their brain when doing such tasks, and therefore they’re not engaging deeply with mathematics while doing it. - Tasks must also
**hold together coherently**across multiple time scales. I try to help students see the connections between tasks they work on during one class session, or on one homework set. I also try to help students see the connections between tasks from September and December. This means that I have to create tasks that honestly support making such connections. One of the most wonderful things about mathematics is how deeply interconnected it is; designing tasks that help students see those connections is a way I can show them the wonder of a subject that sometimes looks quite dry from the outside.

- There’s a lot to unpack here. First, in a math class, we need to be learning math, so our tasks must be
**Students collaboratively process mathematical ideas.**- This is not just a logistical statement about what happens in class on a given day; it is a statement about the general process of learning. To me, this means that if I am not providing time in class on a given day for students to collaboratively process mathematical ideas, then I am not providing them time to learn.
- Four students can be sitting at the same table working on the same task at the same time without
**collaborating**. So, tasks must be**groupworthy**; the physical space must support collaboration; and I have to help students learn to work together as equals. - I like that this statement is agnostic as to the source of mathematical ideas. It’s okay for me to introduce an idea I want students to think about — as long as I then give them room to process it collaboratively. Excellent teaching means moving responsively along a continuum between telling and discovery.
- This pillar implies a particular kind of caring and openness in the community of the classroom. Ideas are valued and examined, no matter what: whether an idea comes from a student or an instructor, whether it’s complete or a rough draft, whether it’s ultimately correct or incorrect, we work together to process it and learn from it. I work hard with students to negotiate norms and expectations that foster this kind of classroom community.

**Instructors inquire into student thinking.**- First of all, this is my
**favorite**part of my job. - This is also a
**key**part of my job. If I’m out to help students improve their understanding of mathematics, then I must diagnose their current understanding. So, if I am not providing room in class for students to express their thinking, then I have no hope of understanding it, let alone of helping them improve it. - Inquiring into student thinking helps students sharpen their thinking. Making students explain their thinking to me (and to other students) helps them see what they understand, solidify their understanding, and identify the precise things they’re still having trouble with.
- This pillar helps inform my assessment philosophy: Assessment isn’t about giving points, it’s about understanding student thinking. So, I approach students’ work in the same way that I approach a conversation with that student: as an opportunity to understand their thinking and to help them sharpen it.

- First of all, this is my
**Instructors foster equity in their design and facilitation choices.**- I need to
**design**equitable course experiences. For instance, I use open educational resources (OERs) whenever I possibly can, to help lessen the financial burden of education; I incorporate universal design principles to allow students multiple opportunities and multiple pathways to develop and demonstrate competence; and I carefully craft syllabi that are understandable and navigable. - I need to
**facilitate**equitable course experiences**in the moment**. For instance, I help students (especially those from minoritized backgrounds) develop their mathematical identity and power by assigning competence (another reason to inquire into student thinking!); I work with groups to ensure that every student’s voice is heard; and I work seriously with campus disability resource centers to modify courses in order to support students with disabilities. - I need to know what
**equity**means in the first place. (Gutierrez’s framework has been influential on my thinking so far.) I need to interrogate and address my own unconscious biases. I need to explore ways to dismantle oppressive systems, even when they are ones that have benefitted me. - I’m a proud gay man, and I’m out in the classroom, because I want to help LGBTQ+ students see that there are people like them who have fulfilling professional lives. I didn’t see many people like me when I was in college; more visibility of this sort would have been a great boon to me during my college years.
- I’ve given a few examples of things that are currently in my toolbox, but it’s extremely important to me to continue to learn new ways to recognize and address inequity in my classroom.

- I need to