Back in the day (most commenters say about 20 to 30 years ago), overheating ovens were a real concern. In some cases, the hot ovens were a fire risk; in others, the oven would automatically shut off when they got too hot, interrupting the broiling process. Leaving the oven door ajar allowed enough air to cycle in without dramatically reducing the temperature—so the broiler remained on, without the risk of smoking or or overheating.
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Logical as that sounds, broiling (and overheating) actually play out differently in different ovens, depending on the year, brand, and type (electric, gas, free-standing, slide-in, etc.). Many modern ovens have certain features that prevent overheating, smoking, or inconsistent temperatures. For example, I checked a gas Viking oven manual, which specifically advised:
"Close the door... A built-in smoke "eliminator" in the top of the oven helps reduce smoke and odors."
Food52er mainecook61 checked her new GE oven's manual, too, which she says was "clear about broiling with the oven door closed. I have to remind myself, because my former (old) oven required an open door." Inpatskitchen says that her broiler flame actually goes out if the door is left open for 30 seconds or so.
As a rule, gas ovens advise closing the door while broiling, while electric suggest to leave ajar. But there are many exceptions—Eric says the digital display on his his GE electric oven quite literally orders him to "CLOSE DOOR" before the broiler turns on.
Given the extent of variables when it comes to ovens, the best advice is to check your manual. You may feel like you're overruling your mother, but you can assure her that, circa 1970, she was probably right.
If you don't have your manual handy, follow susan g's lead and look for context clues. "My mother's oven was built so that the oven door had a point where it was meant to stay open," she says. "My oven door won't stay open unless I hold it—so they have made their intentions clear."