ZW/12

Field notes

What the work is teaching me.

Short reflections on research, building, and the classroom.

Mathematics education reflection

What Happens Before "Notice That"?

While reviewing for MATH 351, I reached a point where I was thoroughly fed up with the phrase “notice that” in mathematical proofs.

“Notice that we can choose ϵ=(L-a)/2.”

Looking at that choice, I knew it was positive, and I could follow the ensuing deduction to verify that it indeed worked. Yet I still wanted to ask: Why that specific choice? How did you come up with it? Why were you able to “notice” it, while I could only understand it after you wrote it down?

Back then, I felt the instructor’s observational power was simply superhuman.

Later, I gradually realized that—at least in these kinds of proofs—I needed to make a mental move I hadn’t been conscious of before: starting from the desired conclusion, working backward to find conditions sufficient to make it hold, and then returning to the hypotheses to verify that those conditions could be met. While a formal proof is written in a neat, forward-logical order, the actual process of discovery often involves pacing back and forth many times.

This realization brought immense relief. I finally understood that I could actively switch the direction of my thinking; I didn’t have to keep pushing blindly forward from the given premises, waiting for some magical choice to strike.

Immediately, though, another thought followed: What about the students who haven’t realized this yet?

If a learnable strategy is never explicitly articulated yet silently becomes a prerequisite for keeping up with the course, students who get stuck might have no idea what they are missing. They see someone else writing down a proof effortlessly, while they can’t even fathom the first step. In that vacuum, “I don’t yet know how to search for this proof” easily morphs into “Maybe I’m just not cut out for math.”

I cannot extrapolate why other students struggle based on my experience alone. But I have begun to wonder: do we sometimes rush to mistake unfamiliarity with hidden scratch work for a fundamental lack of ability?

My MATH 351 class used Inquiry-Based Learning (IBL). There is real value in letting students experiment, discuss, and discover mathematics for themselves. Nor do I want every problem pre-packaged into a mechanical checklist to execute blindly. Still, I wish classrooms left more room for the process of discovery itself: showcasing rough drafts, explaining failed attempts, or even saying outright, “Pushing forward from here, I’m stuck for the moment—so let’s see what the conclusion actually demands.”

Everything that happens before “notice that” is mathematics worth teaching, too.

Discussing this experience with AI helped me frame my confusion into a more concrete question: How much of an expert’s proof-search process is made visible in the classroom? This question isn’t unique to any single teaching pedagogy. Whether in a large lecture or small inquiry groups, students may still need guidance to understand how a particular choice was born.

This also sparked an idea for a small project: collecting concrete moments from different students in the classroom.

In which class, and at which step, did they start falling behind? When did their attention break? When did the thought “maybe I lack talent” first cross their mind? What later helped them grasp the concept anew, or gave them the resolve to keep trying?

These field notes might be scattered and fragmented. One person might stumble over notation; another might wonder why an auxiliary variable was suddenly constructed out of thin air; someone else might follow every individual step yet remain entirely lost about where the overall proof is heading. These fragments may not coalesce into a single, tidy pattern, and students’ reflections at the time might not pinpoint the true root cause of their difficulties.

Even so, I hope these glimpses can serve as a starting point for deeper understanding. By juxtaposing them with the course material, the instructor’s explanations, and the students’ subsequent learning trajectories, perhaps we can diagnose with greater precision: Where exactly did a student lose the thread, and what support do they need to reconnect it?

I am not yet sure if I have the time or resources to carry out this project. Right now, it is merely an idea that sprouted from the frustrations of exam review.

Yet it has already changed how I see learning obstacles. I want to ask: before a student decides they are “not smart enough,” has anyone helped them pinpoint the exact hurdle in front of them? Has anyone told them that the polished proof before their eyes might well be the product of hesitation, false starts, and repeated changes in direction?

I hope mathematics education can help more students demystify their struggles and find a way forward. And I would like to begin with the very moments that are so easily glossed over with a casual “notice that.”

This piece grew out of my reflections during MATH 351 review and a discussion with AI. The project described remains in the conceptual stage.

Digital folklore reflection

The meme whose author disappeared

A recent attempt to trace the origin of a viral video became my introduction to digital folklore. I could trace the clip to a 2016 repost, but not to the person who filmed it or an original upload. The video remained culturally visible even as its provenance disappeared.

We often say that the internet never forgets. But a digital artifact can survive while its creator, context, and transmission history become difficult—or impossible—to recover. Reposts preserve content while stripping away metadata; deletions and platform changes weaken the archival record; repeated edits can make the earliest surviving version look like the original.

That gap changed the questions I wanted to ask. Rather than stopping at what a meme means, I want to understand how it acquired that meaning: who created it, which communities and institutions circulated it, how audiences reinterpreted it, and what parts of its history were lost along the way.

This is where my interests in computing education and digital folklore meet. Both are human-centered inquiries into how technology mediates knowledge, meaning, and memory—and both can benefit from technical methods without treating technology itself as the final subject.

Research reflection

What two weeks taught me about research

Two weeks is not enough to produce a mature research project. It is enough, however, to experience the rhythm of research: learning unfamiliar foundations, forming questions, testing novelty against the literature, running experiments, and deciding what the evidence actually supports.

When our group began, we did not all share the same background. I created shared onboarding materials and a workshop schedule so that everyone could begin from a common foundation. Giving the team a common starting point made better questions possible.

We tested early questions against the literature and the available evidence, setting aside directions that were not novel or well supported. By the end, we did not have a publication. We had something more appropriate for two weeks: a sharper question worth pursuing, a team that had learned how to work together, and a clearer understanding of what research progress can look like.

What I remember most are the small acts behind that progress—handling logistics, sharing resources, and helping teammates protect time for focused work. The workshop taught me that research is not only about results. It is also about building the conditions in which careful work can happen.

Teaching reflection
Tim, Dr. Edgar, and Gary smiling together after teaching AI Ain’t Magic
AI Ain’t Magic, left to right: Tim, Dr. Edgar, and me.
A whiteboard lesson tracing gradient descent and backpropagation through a neural network
Gradient descent and backpropagation, worked through step by step.
A whiteboard connecting I, Robot, Pascal’s Wager, and decisions with potentially infinite stakes
Connecting Pascal’s Wager and I, Robot in an AI ethics lesson.

Four years later, I became the teacher

AI Ain’t Magic was my first teaching experience at Michigan, and it was unlike the recitations and office hours I had led before. Instead of supporting one semester-long course, I helped introduce machine learning to high school students with no prior experience in the field.

I taught mini-lectures on gradient descent, probability, AI ethics, and philosophy. I also led a mock trial asking whether an AI image-generation company had violated the rights of human artists. Moving between mathematics, humanities, and social questions made one lesson especially clear: AI education is inherently interdisciplinary.

Preparing those lessons gave me a more complete view of material I thought I already understood. Four years earlier, I had been a high school student learning R in a data-science summer program. Now I was the person standing at the front of a university classroom. That full-circle moment reminded me that teaching is not the final step after learning—it is one of the ways learning continues.

Project reflection

Designing for learning, not just output

Working in computing education has made me more attentive to the difference between building a technically capable system and building something that genuinely supports learning.

The questions I return to are intentionally broad: Does a tool help people understand and reflect? Does it preserve learner agency? Can educators interpret what it provides and use it responsibly?

The project is ongoing, so I am keeping its mechanisms and findings private. What I can share is the principle guiding my contribution: educational technology should make reasoning easier to engage with, not less visible.

“The internet can preserve an artifact while losing its history.”