The phone - with its ability to stream episodes of Dora the Explorer - is a godsend, Mullaney says.
Attempts at going out without whipping out the gadget have been disastrous, the Denver mom says. Her curious, independent toddler gets into everything. Salt shakers are fiddled with, drinks are spilled.
"She'll color for a little bit or talk with us for a little bit, but it's short-lived," Mullaney says. "It's miserable because all she wants to do is get out."
With the iPhone, however, Tatum sits quietly in the booth while her parents get to enjoy a meal.
Mullaney, a marketing manager for a technology company, sometimes wishes they could do without the phone because she doesn't want people to think they're using technology to shut their child up, but she also doesn't want to give up going out.
"Sometimes you gotta do what you gotta do," she says.
Mullaney is in good company. About 40 percent of 2- to 4-year-olds (and 10 percent of kids younger than that) have used a smartphone, tablet or video iPod, according to a new study by the nonprofit group Common Sense Media. Roughly 1 in 5 parents surveyed said they give their children these devices to keep them occupied while running errands.
There are thousands of apps targeted specifically to babies and toddlers - interactive games that name body parts, for example, or sing nursery rhymes. It has become commonplace to see little ones flicking through photos on their parents' phones during church or playing games on a tablet during a bus, train or plane ride. Parents of newborns rave about an app that plays white noise, a womb-like whoosh that lulls screaming babies to sleep.
In fact, toymaker Fisher Price has just released a new hard case for the iPhone and iPod touch, framed by a colorful rattle, which allows babies to play while promising protection from "dribbles, drool and unwanted call-making."
Denise Thevenot acknowledges that some people would look askance at the idea of giving a child a $600 device to play with - she had the same concerns initially. Then she discovered the sheer potential.
"The iPad is movies, books and games all wrapped in one nice package," says Thevenot, who works in the New Orleans tourism industry. The iPad, she says, keeps her 3-year-old son Frankie busy for hours. And, when needed, taking it away "is the greatest punishment. ... He loves it that much."
Kaamna Bhojwani-Dhawan is an unapologetic proponent of the trend.
"If you're raising children, you've got to raise them with the times," says Bhojwani-Dhawan, who lives in Silicon Valley and founded the family travel website Momaboard.com. "If adults are going all digital, how can we expect children to be left behind?"
Her 2 1/2-year-old, Karam, loves the GoodieWords app, which explains complex concepts like "shadow" and "electricity." Other favorites are a memory matching game with farm animals and a drawing program.
Bhojwani-Dhawan points out that Karam also has books, crayons and Legos. "It's not replacing any of these things; it's one more thing he's getting exposed to," she says.
Experts say balance is key.
"It's really important that children have a variety of tools to learn from. Technology gadgets can be one of those tools, but they shouldn't dominate, especially when we're talking about very young children," says Cheryl Rode, a clinical psychologist at the San Diego Center for Children, a nonprofit that provides mental health services.
"If kids are isolating themselves or if it's narrowing their range of interest in things - everything else is boring - those are big red flags," Rode says. "You want them to have the ability to find lots of different ways to engage themselves."
For public relations consultant Stacey Stark, one red flag was seeing her 1 1/2-year-old cry if she wasn't allowed to hold Stark's iPhone. Little Amalia has dropped the phone, leaving it with a small crack on the back. She has also called a colleague of Stark's and almost shot off an email to a client.
For all those reasons, Stark and her husband have started to cut back on how much they let Amalia and 4-year-old Cecelia use their phones and tablets.
"It became an issue. We're trying to make it go away," the Milwaukee mom says. "It was easy for it to become a crutch."
Since scaling back, Stark says, she has seen her daughters engage in more imaginative play. Still, there is a positive side to the technology, Stark says. She thinks Montessori reading and spelling apps have accelerated her older daughter's learning in those areas. "But," she adds, "it's such a delicate balance."
Wake Forest University psychology professor Deborah Best, who specializes in early childhood, agrees that children can benefit from programs that are age-appropriate and designed for learning.
But "interacting with devices certainly does not replace one-on-one, face-to-face interaction between children and parents, or children and peers," Best says. Those interactions, she says, help children learn such skills as reading emotions from facial expressions and taking turns in conversations.
Joan McCoy, a bookstore owner and grandmother of five in Seattle, worries that this new generation will lack some of those social skills.
When her son and daughter-in-law get together with other parents and their kids, they give the children cellphones to play with, or the children bring along toy computers. "There is absolutely no conversation among them or with their parents. They are glued to the machine," McCoy says.
It's a different story when the youngsters, ages 2 through 7, are out with their grandmother. McCoy brings along books, sometimes ones with only pictures, and asks the kids what they think is going on and what they would do in a similar situation.
"They just talk and they're excited and they're engaged," McCoy says. "They never ask for my cellphone, which is amazing because when we go with the parents, that's the first thing they ask for."
McCoy acknowledges she has the luxury of being a grandparent and having the time to do these things. "It's harder. It takes more discipline, it takes more time, and it requires interacting with the child as opposed to the child being entertained on their own," she says.
Eileen Wolter, a writer in Summit, N.J., readily admits to taking the easier path with her 3- and 6-year-old sons: "I'm buying my kids' silence with an expensive toy."
When her in-laws get together for a family meal, six iPhones get passed to six children. The adults talk while the kids play, their contribution to the discussion typically limited to announcing they have cleared another level on a game. When that happens, Wolter starts to think, "Eek!"
But then she says to herself, "Yeah, but we had a nice dinner."