As much as I hate to say it, I doubt your mom will believe us either. Since this is a fish forum, she'll probably write us off as fanatics (Like I do with PETA). ^_^;
But here are my two cents anyway. :D
My personal min is 1 gallon. While on campus I had a 1 gallon tank (the corner tank you get from petco for $12 i think), and could fit a heater, thermometer, filter, small castle, and 1 tiny fake plant with a betta. Needless to say any betta in that set up didn't have much swimming room, but the water was clean water at the proper temperature. My water change schedule was 2 50% and 1 100% water change a week.
Average life expectance for a fish in my care in a 1 gallon tank is 2 - 2 1/2 years. Some people will argue that 2 years is too short since the average life span of a betta is 3-5 years old. But, since I have always brought my fish at pet stores, I could have come home with 8 - 10 month old fish without knowing the difference.
My current set up is a 5 gallon tank. It has a filter (that I had to baffle the lifing snot out of -_-), heater, thermometer, two hiding places (one big one small), some live plants (new addition), tons of swim room. Unlike a 1 gallon set up, a 5 gallon set up can be cycled. Once a tank is cycled it is basically on autopilot. My current water change schedule is 25% once a week with sand surface cleaning and 50% once a month with sand surface cleaning.
I can't guess if that increase life spam yet since both Luigi and Snowy lived most of their lives in 1 gallon set ups. But I can say that they were much more active had their personalities showed a bit more. They definately used the extra space once I upgraded their tanks. :)
So I go either way. I have had healthy fish in 1 gallon setups in the past without shorting there lives drastically, but it was difficult to keep up the water quality. I definately had random ammonia spikes in my old 1 gallon setup and still have a habit of using a lidded bucket to hold "emergency water" to this day. Not that I need it with a cycled tank that is very stable and doesn't give me the same fluctuations I dealt with in the past.
So, while your mom is technically right about a betta surviving in a bowl (if it is at least 1 gallon), I don't think she realizes how much work goes into keeping a betta alive in 1 gallon of water. If your mom doesn't want to be bothered with daily water monitoring and water changes 3 times a week, she should get at least a 2.5 gallon tank. She wouldn't be able to cycle it like I cycled my 5 gallon tanks, but its parameters would be more stable than a 1 gallon bowl and need less hands on maintinance.
In fact, if the bowl your mom got holds 2-3 gallons of water, she's already on the right track by accident. :)
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