Tom Pashut reached for the collar on his shy Cavalier puppy, Lucy, while she watched — eyes wide as dinner plates — as his other hand moved to the fanny pack on his hip. Pashut retrieved some diced hot dog, showing Lucy that a hold on her collar means good things follow.

Standing in a classroom at SF Puppy Prep, trainer Mac McNamara nodded in approval. “You just need to play this game a thousand times and she’ll be good,” McNamara said.

This is puppy kindergarten. When Lucy graduates, it will only be the start of her education: Single dog-dad Pashut has already forked over $560 for Lucy’s enrollment in puppy junior high and high school at Puppy Prep. And in the unlikely event she flunks out of school, there are boot camps around the Bay Area where dog owners send their barkers, biters and whiners for training.

The last census showed San Francisco has the smallest percentage of children of any major U.S. city, with about 116,000 residents under the age of 18. There are slightly more dogs, between 120,000 and 150,000, according to estimates by San Francisco Animal Care and Control.

As it becomes increasingly difficult for people to afford having a family in San Francisco, humans are giving their dogs a formal education that pets living outside cities can only imagine.

Here, dog owners enroll their four-legged children in manners classes so advanced they have prerequisites. Some organizations offer extracurricular activities such as “doga” (dog yoga) or circus school. Owners may spend $200 for a beginner’s class at San Francisco SPCA, or shell out $2,450 for the “magna cum laude package” at Puppy Prep, which includes a month of day care, puppy socials and parent-teacher conferences. (Unlike elite schools for human learners, no bribes are necessary for admission, however.)

Pashut, an accountant who takes Lucy to work at Uber, had one month to ready for her arrival from the breeder. He found a veternarian, bought toys and food, and signed Lucy up for Saturday morning kindergarten classes. His goals are for Lucy to come when called and to ignore trash when she hears “leave it.”

“On the streets of San Francisco, you never know what you can find,” he said.

After Jed and Simone Bargen had to give away their last rescue dog, a scaredy-cat who turned aggressive toward people, they ended up with Cooper, an overly friendly mutt. Squatting on the floor at Puppy Prep, Jed closed his hand over a piece of dry food as Cooper tried to nudge it open with his nose. As soon as Cooper turned away, Jed made a clicking noise to mark the behavior and fed Cooper some freeze-fried salmon, laying the groundwork for the “leave it” command.

“We save the stinky stuff for when we really need him to pay attention,” said Simone, who runs a small IT business with Jed.

Cooper is also a student at Puppy Prep’s day school, a popular program that lets owners drop off their puppies before work for seven hours of play and training. The goal is to expose puppies to as many San Francisco sights and sounds as possible, helping to mold them into confident adults. They might meet a toddler, play with vacuums or watch staff zoom by on skateboards.

McNamara, a trainer and owner of Puppy Prep since 2017, said they even introduce puppies to shopping carts filled with jingling cans to prevent fears of homeless people.

“We’d like him to coexist with the people, the garbage trucks and the neighbors above us,” Simone said after class, as Cooper lay on a beach towel he’s trained to relax on at restaurants.

In the suburbs, people with feisty pets have the luxury of keeping them in a yard. “It’s hard to ignore a dog behavior problem when you live in a city,” said Camilla Gray-Nelson, who has been training dogs on her family’s ranch in Petaluma for 30 years. “You can’t throw them outside.”

McNamara added that most suburbanites send their dogs to school only if their breeders require it. “Here, I think people say, ‘This is my first child. I want to do it right,’” she said.

As the city’s canine population ticks up, more pet owners are seeking their doggie doctorates. Enrollment in classes like puppy manners, “shy dog seminar” and “freestyle heelwork” at San Francisco SPCA rose to about 6,000 registrations last year, up 200 percent from 2014.

But money can’t buy a good boy, said Gray-Nelson, the trainer who converted her family’s dairy farm into Dairydell Canine, a finishing school for dogs.

Set on the sloping hills of Petaluma, it enrolls more than 250 dogs a year to help them become well-mannered. Boarding programs last from one to four weeks and cost up to $5,400, while many “parents” come back on weekends for maintenance classes.

Best in class Dairydell Canine: 707-762-6111, www.dairydell.com. Petaluma finishing school for dogs of all ages. Koru K9: 415-583-5412, www.koruk9.com. Bay Area private training and last-resort rehab. SF Puppy Prep:www.sfpuppyprep.com. S.F. school with basic classes for a well-rounded puppy. SF SPCA: 415-554-3000, www.sfspca.org. S.F. workshops for shy and overreactive dogs. Refined K9: 415-980-4200, refinedk9.com. S.F. boot camps for distressed dogs.

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“Affluent clients — some of them, not all — seem to imagine that if they pay enough money, money is the answer. ‘I will get a perfect dog,’” Gray-Nelson said. “Our constant challenge is to let people know obedience is about leadership, and leadership isn’t about money.”

Pashut said the hundreds of dollars he has spent on Lucy’s training isn’t exorbitant when he divides the cost over the years of her life. He thinks about the German shepherd he had growing up. The dog wasn’t around other dogs or people much and became overly protective of the family. After she bit a passing dog, the city impounded her for a brief time.

“When I realized that investing in training is good for both the human owners and the dogs, I promised myself to never let what happened to our family dog happen again,” Pashut said.

At the end of class on Saturday, the puppies practiced coming when called — two at a time in front of the group. At 14 weeks, Lucy weighs all of 7 pounds and trembles near other dogs, so McNamara suggested she go alone.

Noticing all eyes trained on her, nerves struck. Lucy moved to the wall as Pashut tried to cajole her closer with pieces of string cheese.

“Just an inch,” McNamara said softly. Pashut closed the gap between them.

Lucy lifted a front paw, then the other, to take two steps forward. The class let out a cheer as she nibbled on her reward. Baby steps.

This story has been updated to reflect the fact that some Dairydell maintenance classes are covered by the program fee.

Melia Russell is a San Francisco Chronicle staff writer. Email: melia.russell@sfchronicle.com Twitter: @meliarobin