Why do we have them? How did they become so engrained in suburban life? Were they always such a status symbol?
I guess this is my obligatory disclaimer that I don't specialize in front lawn history and hope that if my answer doesn't suffice in answering your questions, hopefully any corrections or other posters can give you a more kick-grass historical context.
The history of lawn development in a broader sense is long and deeply wrapped up in so many of the cultural debates about how nature and art can/should/will (or not) intersect across Western Europe and North America, and I’m only most familiar with later 18th-19th century British sources in that context from Burke to Ruskin; luckily for me, since you seem to have the quintessential golf green suburban lawn in mind, I’ll narrow my answer down to the development of the classic post-World War II American front lawn, even though I'll probably still write too much. I guess a tl;dr for this argument could be that we have front lawns because they are convenient, excellent symbols of a singular upper-middle-class, generally (although not always of course) WASP American culture; they descend from the subjects of those aforementioned nature/art debates, British gardens and landscapes, which themselves often tried to capture a pastoral ideal of aesthetic and social harmony, and the shifting amounts of time, labor, money, and expectations of appearance for traditional front lawns solidified them as an accessible symbol of the prosperous, virtuous American citizen post-World War II.
Cultivated lawn spaces emerged in England around the 18th century, and at the time usually entailed large, mowed garden land made to look artificially and aesthetically pleasing and owned usually by the gentry - see William Marlow’s “A view of the lake and island at Kew, seen from the lawn” to get a better idea of what these lawns looked like. Wealthy Americans in the 18th century often read about or got to see these developing landscapes, as many were popular tourist destinations, and they usually copied the picturesque English appearances of these lawn spaces, but for the most part, lawn spaces did not look like or function as modern front lawns do now, as Marlow’s illustration shows, and most lawns in 18th to mid-19th century America functioned as the commons and did not often begin and end at the “front” of the property (but it is interesting to note that one of the first uses of the word “lawn” in the sense I’ve described above comes from Philip Miller’s The Gardener’s Dictionary, where he advises, “As to the Situation of a Lawn, it will be best in the Front of the House”).
Around the 1860s, front lawns for the private homes of upper class Americans became possible through a combination of better technology to maintain them (the improvement of cylindrical lawnmowers and irrigation devices, for instance), the increasing popularity of sports for leisure that often required lawn space (golfing as a lawn sport will be especially important in popularizing the suburban front lawn later), and a cultural reaction against urbanization hoping to separate the middle and upper classes from their increasingly close working class counterparts in the cities (Andrew Jackson Downing and Frederick Law Olmstead are often cited as the most important landscape designers who foreshadow the creation of suburbs, secluded from harsh city life and meant to uphold an ideal communal life for elites). Still, very few American people in the 19th century had any green space around their homes that wasn’t farmland or maybe a personal garden; lawns required effort and resources that made lawn keeping impractical for most middle class Americans, so that still only wealthier landowners could afford the equipment and workers - lawn mower trade cards like this one can give you an idea of just how large these front lawns are, and this advertisement in particular implies the earlier problems of middle-class women, as the ones usually expected to be interested in gardening and lawn care, when dealing with the too-heavy machines.
Moving into the early twentieth century, the impulse on the part of the growing middle class gradually began to take up the ideas proposed by landscape designers like Downing and Olmstead, and organizations like the Garden Club of America (GCA) and the United States Golf Association (USGA) helped promote the significance of the front yard as a unifying symbol for civic-minded middle-class Americans who wanted to make their community as beautiful as possible in a collective effort. The GCA gave lawn care advice to all GCA members nationwide and sponsored lawn care competitions, and the USGA, as golf became the new rage amongst wealthy Americans in the early 20th century, got help from the USDA in developing sturdier turf that could survive and thrive in various climates - golf greens would provide an aesthetic model that most Americans could use then when creating the classic front lawn look. Magazines in the 1920s like Nature Magazine, American Home, and House and Garden continued to proliferate the luxurious appearance and function of the front lawn, and even though the technology for lawn care was still not adequate enough to make front lawns a practical reality, advertisements for lawns reached even lower-class Americans by this time. Really, most middle-class suburban homes had front lawns by the 1930s, but the lawn care industry, revolving around fertilizers/lawnmowers/irrigation systems/grass planting, had supply and technological problems during the Great Depression and World War II.
These problems abated with the economic boom following the war, and as the real estate business of America revolutionized around creating zoned suburban sprawls for war veterans, their growing families, and their healthy supplies of cash to live, lawn care technology finally became accessible and practical enough to afford and use. Front lawns became the verdant green “carpets” we think of today thanks to voluminous advertisements and often direct interventions from housing commissions hoping to maintain pleasing, uniform neighborhoods so they could sell more houses, although they often sincerely emphasized that lawn maintenance was necessary for harmony between an entire suburban neighborhood hoping to look good. Advertisers and literature began to associate front lawns with the nuclear family, with many lawn mower ads emphasizing that mom/dad/kids could use their mower together, and as a necessary masculine activity - in this way, socioeconomic forces worked to make the front lawn an accessible way for the middle class to emphasize their identities as “democratic,” patriarchal and family-oriented in their dedication to their larger community, all while being able to conspicuously display their wealth in a way that those living in plateauing urban areas, increasingly inhabited by African-Americans and working class immigrants, could not.
My information generally comes from Lawn People: How Grasses, Weeds, and Chemicals Make Us Who We Are, The Lawn: A History of an American Obsession, and Crabgrass Frontier: The Suburbanization of the United States. The cultivation and development of different types of grass provides a lot of context for how the front lawn came to be, but I personally don't have much to add to that angle unfortunately.
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