Volume 4, Entry 33: Metáfora

Comparisons without like, as, or comparisons

My first lesson of AP Calculus AB opens in the same place every year.

The topic on day one is limits, but I don’t actually begin there. As a broad nod to the course as a whole, I instead talk about mathematical representation.

“This course,” I say of Calculus AB, “is to get you analyzing relationships in different forms with the tools of calculus we develop. Let’s establish those three forms first.”

I then offer a brief reference to the numeric, graphical, and algebraic representations we trade in throughout the curriculum. Although I merely note what these consist of here—tables of values, connected dots, symbolic equations—minutes later, we are analyzing a rational function’s behavior using the complete trio.

In both this first segment and what follows, my lesson is effectively the same as it has been since I first rewrote it in 2012. Yes, I’ve significantly refined my language to emphasize the “connected closeness” concept, added flourishes of digital artistry, and snuck some linearity into a graph-based exercise, but the original skeleton remains present. Everything is more polished and flows much better—even with me reaching for wordplay every other sentence—but the ingredients have not changed.

At least, they hadn’t until I added a new element last year. In hopes of decreasing limits’ abstract awkwardness and previewing my penchant for goofy digital storytelling, I made a PowerPoint mini-movie to transition from background information into actual problems. The mini-movie revolves around Steve and I making plans to meet up at Walmart and watching those plans transpire in three very different ways.

I’ll not bore you with the specifics, but it’s silly while still focused. The three limits we compute on the next page match those three Walmart scenarios, which lends the media greater weight. Students can always calculate those answers, but this excursion garners laughs while strengthening their comfort with what limits actually mean.

Despite enjoying the story, I initially cut it this year to conserve time, but I pivoted back to it when my first-period class was stingy with laughter. I ended up using it three times across the day with more than 100 students.

During the third and final time, an amusing thought occurred to me. As we wrapped up the subsequent problem, I seized on this realization and returned to page one of the notes.

“I suppose I lied,” I announced to my class. “We actually represent relationships four ways in here.”

With an intentional disregard for my original well-spaced list, I scribbled out a crude four and wrote “Metaphorical” in the corner. This prompted scattered laughs from several students who assumed I was kidding around.

I appreciated the mirth, but I wasn’t joking. It made sense to include that fourth option. Metaphors are a huge part of how I instruct. They help me make sense of confusing concepts and tricky techniques every day. Figurative language is essential equipment in my teacher’s tool belt.

My deep comprehension of math must transfer into their brains, so I offer metaphors to bridge our understanding. Just this week, I’ve chased that Walmart analogy with ones involving hamster metamorphosis, packed pizza parties, and smashed sandwiches; last night, I even got mileage tying conjugate multiplication to a Harry Houdini stunt. Every comparison I make doesn’t land, mind you, but each one that does inches my students and me toward common ground. It pulls them closer to my multi-dimensional understanding.

It isn’t just an academic thing, though. When I’m stuck on some idea, I reflect on analogous situations. With luck, grappling with the metaphor will inspire insight into the original. Weird though it might sound, this works pretty well for me. Comparisons get me out of my head and moving forward. I become a tourist sliding into an Uber; I become Harry and Hermione hopping onto a hippogriff. The comparison doesn’t engineer my escape directly, but I hitch a ride on it toward progress all the same.

Metaphors are how I help students make sense of mathematics, and they are how I help myself make sense of the world. Bar none, they might very well be my go-to strategy.

Indeed: when I’m perplexed by a pernicious problem, the first place I turn is to metaphors.

*****

NAIL HOUSES

If you’re over by Target and need to mail something, there’s a post office nearby. It’s close to the building that used to be Orchard Hardware, which is now a cosmetics place. Actually, it’s really close: I could play catch with someone between the two locations, rebuilt back and all.

If you‘ve ever tried to reach the post office directly from the Target shopping center while driving, though, you’ll have discovered you can’t. Indeed: there is no access from the big Target area into the one with the post office and Pizza Guys. There’s only one entrance in and out of that tiny strip mall, and it’s found on Bruceville itself. You have to drive onto Bruceville for several feet to get inside. It’s bizarre. It seems inexplicable.

It isn’t, though. That short strip has been there fifteen years now, so you’d be forgiven for forgetting what was there before. If you frequented Hometown Buffet, you drove past something odd sitting there before Nathan’s Chinese Cuisine and the James Mark Salon ever did. What was it?

A house.

I bet those two words are all you need to conceive of the likely story. No research on my part has been done regarding the specific property, but I’m not sure that’s necessary because houses in bizarre locations like that are not unheard of.

Most commonly referred to as nail houses, these homes belong to individuals who refuse to sell to developers. They are thorns in the side of commercial real estate brokers; they are holdout homeowners whose property cannot be bought.

That term, nail house, draws from those pesky hangnails you get if you’re not careful while trimming. Hangnails jut out and deliver pain to your thumb. These houses jut out and deliver aesthetic, if not financial, pain. Their tenants orchestrate that.

The movie Up features popular culture’s most prominent nail house. Its owner, the cantankerous widower Carl, resists developers’ dollars, refusing to sell the home that housed his long life with Ellie. Construction begins anyway: metal beams stretch skyward as bulldozers and cranes rumble into action in all directions. Everyone wants Carl gone—not only does his house hold up the project, but he’s also abrasive, disruptive, and assaults members of the team—but yet he holds out. There’s no choice but to build around his lot until something changes.

While the story of Up sees Carl fasten balloons to his home to fly away, nail houses typically go nowhere. They just stay. A quick image search reveals many such homes and buildings that defiantly rebuff progress. My favorite is undoubtedly the Chinese apartment parked in the middle of a fully-paved freeway. You can almost hear the snarling sigh of a city planner when you view it.

While I’ve never run into one so egregiously disruptive, I understand the romantic dimension of holding out. Standing up to greedy corporations or defying ruthless government seizures screams of heroism. It’s gunning down the villain in a Western; it’s bellowing “This is my turf” and “You can’t buy me” in the face of foul forces.

But that romantic notion competes for dominance in my mind with a different feeling. While a part of me hears the Braveheart speech while remembering that humble house in the Target shopping center, another part of me feels discomfort with its memory. That area’s inconvenient lack of flow is the owner’s enduring mark on the community: that bold, maybe even admirable, refusal to sell to the original developers has long been forgotten and replaced by annoyance. That’s the legacy of the defiance: people have to leave the shopping center, merge onto a busy street for a few awkward feet, and then pull into this tiny lot to mail a birthday card.

I know I would never hold out for as long as that person did. I couldn’t sit on my porch like Carl, surrounded by the grating cacophony of power tools and industrial machines while a parade of mustachioed men in hard hats glared at me. My home would have been demolished by then. Glimpsing Hometown Buffet and Chevy’s while grabbing the morning paper would do nothing for me.

Never would I want these people forcibly removed or, God forbid, killed for their resistance. Should they eventually cave, I’d expect nothing less than lucrative compensation packages and upgraded residences in exchange. I admire that holdout homeowners can stand their ground—historically, land gets taken by bloody force and false promises, so this is a positive change—but I can admire the principled stance while also rolling my eyes at its eventual outcome.

If I’m leaving sushi and have to ship a package, I growl, remembering the stubborn nail house that laid a concrete moat around that run of suites. Good on them for standing firm as long as they did, but look at the pettiness their holdout got us.

Impressive or not, sometimes sticking around just leaves a mess in its wake.

*****

HELMET FLAPS

Watch youth baseball and MLB games in close proximity, and you will notice many differences. Is it a surprise to note nine-year-olds play at a different level than 29-year-olds? I would hope not.

Let’s talk equipment, though. There’s a definite difference in bats—wood versus aluminum, for instance—but other elements might be more difficult to discern. A decent kids’ glove replicates its professional counterpart quite well, and the uniforms, while much cheaper and of lower quality at local parks than at Yankee Stadium, still pop with the coordinated glow of MLB’s color palette. Even less distinguishable are the hats: while the pros don the fitted hats I favor, kids’ hats look pretty legit. Everything is smaller but otherwise, superficially, the same.

You will only notice a striking difference in one place: the batting helmets. While there are two large ear flaps protecting the cranium of every youth player in the country, only a handful of Major Leaguers head to the plate sporting a pair. It’s one flap or bust for almost every one of the best baseball players in the world, both here and in Japan.

That wasn’t always the case. Until the 1930s¹, batting helmets weren’t a thing at all. Even when Cleveland shortstop Ray Chapman died after taking a pitch to the head in 1920, there was no persuasive push for change. In 1937, rumblings began, but it wasn’t until 1941 that any top league mandated that its players protect their skulls.

The helmets they initially required had zero flaps, by the way. They were glorified plastic hats with a bit of padding in them—great for minimizing the damage of an errant pitch to the skull, but only to the skull. These helmets were prone to flying off while running the bases, exposing heads to dangerous defensive throws as well, but they were better than nothing.

In 1983, the helmet rule changed. From that year forward, every MLB player had to wear at least one ear flap on his helmet. Facing toward the pitcher—the left side for righties, the opposite for lefty swingers—this flap would consist of hard plastic and cushioning as well but, vitally, protect the vulnerable eyes, cheek, and temple from blunt force trauma. Players had been dabbling with this design since the mid-sixties, including several Hall-of-Famers, so this evolution offered safety without wounding one’s reputation.

As I became a baseball fan in the early nineties, I came to see those one-flap helmets as the epitome of cool. The goofy one-size-fits-all cheap ones I wore on the 10U Rockies looked terrible and felt worse; I yearned to wear one flap like Mark McGwire and Barry Bonds did instead.

Watching a baseball game on TV one day, something caught my eye. I didn’t recognize it at first because nobody was talking about it, but I knew something was off.

The hitter in question was Gary Gaetti of the then-California Angels. What made him noteworthy was his helmet.

Gaetti’s helmet had no ear flaps.

Laugh away at the silly little boy I was, but I vicariously panicked for the Angels’ third basemen. I had seen a pitcher take a line drive to the face on the first night I ever watched baseball on TV, and I felt certain I was about to watch Gaetti’s nose get crushed by a Bob Welch fastball. I jumped out of the chair and ran upstairs to get my dad, assuming he too would leap at the emergency, dismounting his exercise bike to phone the Coliseum or something.

When I told him the grave danger I’d just witnessed, he gave me a bewildered look and then chuckled.

“No, no. Gary can wear that,” he said, wiping sweat from his brow. “Gary’s grandfathered in.”

Gaetti made his MLB debut in 1981, two years before officials enacted the one-flap rule. My father’s response referenced the “grandfather clause” of that equipment agreement with the Players’ Association. Although every MLB player had to wear at least one flap from that point forward, officials included an exemption for players who debuted prior to 1983. They alone could continue wearing flapless helmets.

That is to say: the rules didn’t apply to Gary Gaetti because he showed up before they hit the books.

You can be sure I tried to carve out similar grandfather clauses to everything—“I sat down for dinner before you cooked those carrots, so I’m grandfathered out of eating them”—but there was something about the specific one that perplexed me.

Why didn’t Gaetti want to switch?

The rule wasn’t some pointless Byzantine edict. It made sense! The safety of that one flap felt intuitive; it was logical to protect one’s head as much as possible. Why didn’t Gaetti worry about getting hurt? Where was his self-preservation instinct?

Even more on my mind than safety was aesthetics. MLB uniforms represent the apex of fashion. Crisp, clean, and color-coordinated, even my least favorite uniforms still strike me as sharp. It’s hard to look bad wearing any one of them.

Gaetti’s helmet didn’t look bad, per se, but it did make him stick out like a sore thumb. He looked odd; it made his large mop of hair stick out in an unusual manner unique to his flap-free headgear. I was used to one-flap helmets and relished their look; seeing that one oddball lid messed with the image I conceived in my head. Novelty be damned: I wanted Gaetti to be safe and blend in with his peers! But for that pesky grandfather clause, he would have.

I’m happy to report that the man never suffered a severe injury to his noggin from that helmet. I’m also thrilled to report that when he hung up his glove for good a few weeks into the 2000 regular season, the hitter’s no-flap cap followed him out of Fenway Park. With Gaetti’s Baseball Reference page finally complete, that pesky uniform exemption was finally put to pasture. Which is good! The game has evolved past the era of the no-flap helmet. It was time for it to leave. Hell, it was supposed to leave seventeen years before it did. Good riddance.

If that sounds a little harsh, please forgive me for being testy. I hold no ill will toward Gaetti, of course; by all counts, he’s a good dude. But I was happy to see his helmet relegated to archival footage and 1990s Topps factory sets. There was something immensely satisfying about knowing his grandfather clause finally got removed.

Because not everything grandfathered in actually does.

*****

THE FREEZER

When I first injured my back, it took months of pain to truly get help. From March through September, I was hobbled by an invisible neurological knife tearing into my calf, all-day every day. I needed a walker for the first month of school.

Few things helped relieve that pain—there’s a reason I eventually needed surgery—but my original physical therapist suggested I ice it regularly to manage it. At home, that meant withdrawing a fancy ice pack every few hours, but at school, opportunities for that were limited.

So it was that my parents bought me a mini-fridge to put in the classroom. Equipped with a small freezer at the top, the four-foot-tall device allowed me to stash an ice pack at school to mitigate some of the pain mid-day during my lunch.

Since surgery, that fridge has been more traditionally employed. The Pepper Jack cheese I place on my daily quesadilla gets stored there, as do Capri Suns and the occasional student’s lunch or canned beverage. This means, in a total reversal of functional fortune, the freezer that motivated the original purchase usually lies empty. I barely think about it most days.

This past spring, I ended up needing it one time. A student returned from an appointment and brought along McDonald’s. In addition to her food, she brought along a gift for me: a Strawberry Shortcake McFlurry.

I had never had one before, but my initial bites were appealing. Breaking my normally rigid no-snacking policy, I swallowed down about half of the treat and then went to save the rest for later. McFlurries are ice cream, though, so the fridge was out of the question. This job called for a freezer.

There was a problem: my fridge’s freezer is extremely shallow. While the perfect size for flexible ice packs, this small space proves a poor match for upright cups. Without thinking, I set the McFlurry on its side and closed the door, fully intending to finish the thing off after-school or the next day.

Had I put the McFlurry in immediately after receiving it, nothing would have happened. I didn’t: I had eaten several bites across a long enough duration that the dessert phased from solid to liquid. When I placed it on its side, I didn’t consider that the custard was soft enough to spill over the lid’s edge.

It was, and it did. When I opened the freezer to withdraw my McFlurry, I found a puddle of re-frozen ice cream on the icy bed of the freezer. A voice in my head told me to clean it up right then, but I batted that thought away. There was no need! That circle of liquid was a problem, but an objectively tiny one. The freezer was cold; the puddle wouldn’t expand further. There was no urgency to remedy the situation; I could clean it up at any time. I didn’t have to stay late to deal with dairy run-off; I’d get it eventually. Without hesitation, I left that sand dollar-sized circle of pinkish cream in the freezer.

Weeks later, the school year ended. Per instructions from the school, all electronics must be disconnected, so I followed protocol, emptying my refrigerator out, unplugging it, and leaving. I returned once or twice during the break, but I never looked in the fridge. Why would I? There was nothing inside the fridge.

On my first day back, I plugged my fridge back in to prepare for the return of my cheese. Opening up the fridge to survey the place, I found no surprises beyond a few Capri Suns tucked into the back—no big deal.

R’lyeh had advised me to store some calculator batteries in the freezer, though. I hadn’t gotten to fill the set that would be borrowed out yet, so I figured I’d store the bag in my personal freezer for convenience. It would just be more convenient that way. Without a thought in the world, I opened up my freezer to toss the Ziploc bag in.

I stopped before sliding it inside. Because there already was something inside.

In the very center of the freezer sat a dark substance. It was precisely where a tiny puddle of ice cream had been. A tiny puddle of ice cream I had never taken care of. A tiny puddle of ice cream I had left in an unplugged freezer for an entire summer. A tiny puddle of ice cream that had been reborn as a spot of mold.

The freezer is thick enough and sealed well enough that there’s no danger there. The spot remains small and concentrated, too, and turning the freezer back on has arrested any further development from it. But using that freezer is out of the question now; I’ve added two strips of duct tape to its side to discourage any temptation to explore or employ it. Until I can fully and safely address that spot, the freezer is off-limits.

At first, I wanted to blame the McFlurry. Casting aspersions on that dessert effectively shifts responsibility to McDonald’s, and the financial juggernaut can weather the PR hit of a few jabs from me.

Such an allocation of culpability is wrong, though. I know the guilty party here. It’s me, hi, I’m the problem, it’s me. I knew that the ice cream overflow needed to be cleaned up, I knew it, but I decided to address the issue later. It’s a problem, not an urgent problem, I assured myself. Deal with it later.

As a devoted non-procrastinator, I should have done better than that, but I didn’t. Three ounces of a Strawberry Shortcake McFlurry in April have been reborn as a few frozen ounces of mold in August. Neither situation is dire, but the present one is a bigger problem than the prior one. I can’t just wipe it away any longer. By not dealing with it, I let a tiny issue grow into something bigger.

My freezer is sealed shut now. Thick strips of pink duct tape mock me now every day. Letting unwanted things stick around is a dangerous game.

And there’s no one to blame but myself.

*****

It’s taken a month of concentrated effort, but I’ve finally succeeded in cutting back my time spent on Reddit. Instead of several hours in a doom-scrolling fugue state, I’m down to maybe 40 minutes per day. It’s a big change.

My worst days are when something catches my eye. One great post or article reignites my excitement. Perpetually convinced I’m only a few flicks away from my next engaging read, I struggle to stop.

Last week, what caught my eye was a question asked on r/TaylorSwift. A user expressed curiosity about a particular line from Taylor’s hit song “Cruel Summer”, a track released in 2019 but with renewed popularity from its prominent presence on the Eras Tour setlist.

That line in question is the first of the song’s second verse:

Hang your head low in the glow of the vending machine, I'm not dying

The original poster noted the puzzling nature of the line before diving into a thorough thesis. Her investigation eventually culminates in a connection to dating apps and the options they offer. Her perspective has shifted.

“It's not a literal vending machine,” she writes. “It’s a metaphor.”

More than 100 commenters join into the fray from there, and the prevailing interpretation (as told by upvotes) runs counter to OP’s: the vending machine does carry metaphorical value in its tie to choice, but the line specifically references a particular media trope in which characters with hospitalized loved ones search through vending machines for food. They look carefully and thoroughly from the bottom row to the top, caught in an apparent stupor over junk food while life hangs in the balance one room away.

Still, that wasn’t the only explanation extended. Dozens submitted hypotheses and offered opinions before the post receded into the background behind newer arrivals. I never chimed in, but I did find my mind dwelling on it long after I closed the app.

It wasn’t on the lyrics. What stuck with me was the nature of what Taylor had done in “Cruel Summer” and so many other songs. Instead of making a direct comparison for the listener—“The world is a vending machine of possibilities, but I want you to choose me”—she simply presents her image and leaves its relevance unstated. Vending machines have inherent connotations fans can draw from, the rest of the song and her complete catalog both contribute context as well, and hey: a lot of her life is public knowledge. Curious fans can piece together what it all means if they want. Otherwise, they can enjoy a catchy song with distinctive imagery front and center.

I teach math, so I make a living tracing clear connections between steps and ideas. I write accordingly: my goal is to lead readers through my thinking toward a specific conclusion. It’s a guided tour at a thought-provoking museum; everyone takes away something different, but there’s no ambiguity over what each display piece is. I tell you. I identify the intersection points; I bridge the metaphors to their original objects.

Isn’t it fun to ponder Taylor Swift’s lyrics, though? Posts like the vending machine one take me back to AP Lit and breaking Pablo Neruda’s “Soneto XVII” down to its syllables. In a way, that’s a poor comparison, though: Neruda’s poem is metaphoric from the start, but the Chilean wordsmith makes his comparison clear. He is talking about the special nature of his love for some person. The love they share hides; Neruda’s meaning does not.

I certainly got a lot out of that poem back in the day, but I also take a ton from Taylor Swift’s lyrical magic today. I can still recite “Soneto XVII” in its original Español, but my literary analysis at seventeen honestly thrilled me less than puzzling over “Cruel Summer” does today. That makes sense: Taylor doesn’t always say what she means; she makes fans work for it. Every scrap of understanding feels earned.

What is a metaphor without an explicit comparison being made?

To the author, it’s a relief. They can explain what they’re feeling without getting bogged down in the messy, agonizing details.

To a reader or listener, though, it’s different. They don’t know what those things mean. They aren’t psychics; they get left on the outside looking in. That can be infuriating, but it can also be exhilarating.

Because a metaphor without comparison is a mystery.

---

I’m not sure that I have a way to explain this that doesn’t undercut what it is trying to do. I will provide this sliver of context: there exists a Volume 4, Entry 30.5 that I fully wrote, proofed, and created art for. That piece builds around the intersection between a personal conversation, a situation in a friend’s life, and the situation I’m grappling with here. I knew it could never be published when I wrote it; I followed my newsletter process to interrogate the emotional triangle in hopes of gaining traction. What you see here basically washes away all of my narration around these metaphors’ actual meaning. I’m aware this can’t make sense, but it felt so uncomfortable writing in the newsletter style without the ability to share that writing this piece became necessary for my brain to move forward. And I think it finally has: I sketched out an outline of a potential next piece this morning.

This was a good week. Those are nice to run into now and then.

1. https://en.wikipedia.org/wiki/Batting_helmet