Tikki Tikki? No. No.
I certainly don't want to read this to my two sons. Notwithstanding the possible racism, which I have to confess I was mostly unaware of due to my lack of education about other cultures (the illustrations made me think of Japan, but I didn't want to assume that China couldn't have similar art and clothing styles).
The story is this: there are two brothers in "ancient China", where the first and oldest son traditionally has a very long name, and the second son has hardly any name at all because they don't really matter. The elder brother's name is "Tikki Tikki Tembo-no Sa Rembo-chari Bari Ruchi-pip Peri Pembo", which is not Chinese, but the book claims means "The most wonderful thing in the whole wide world," and the youngest is "Chang", which apparently is Chinese, but does not actually mean "Little" or "Nothing". According to the internet, "Chang" means "constant" or "often", but depending on the intonation, can mean a whole lot of things, including "long" and "great" (as in "The Great Wall of China"), so kind of the opposite of "Little." Anyhow, every morning their mother washes clothes at a stream by her house. On the bank of the stream, there's an old well. Which seems a strange place to put a well, since clearly there is water right next to it. Unless the people were concerned (and knowledgeable) about sanitation, in which case they were much more advanced than ancient Europeans. Again, I can't say whether or not this is accurate, as I don't know enough about ancient China, or this mythical place that the author has conjured up which she claims is equivalent. The mother says not to play by the well because they will fall in, and they don't listen to her and Chang falls in. The older brother goes to the mother and says, "Chang fell in the well." Initially, the stream is so loud that she can't hear him, but then he yells at her and the mother says, "That troublesome boy." TTTSRBRPP has to go get "The Old Man with the Ladder" (kind of a sad thing to be known for), who lowers the ladder into the well, climbs in, and brings the boy back out. Chang gets better very quickly. They stay away from the well for several months, but then there's there's a festival and they go to the well to eat their rice cakes, play around, and TTTSRBRPP falls in. Chang goes to his mother and says that Tikki Tikki Tembo-no Sa Rembo-chari Bari Ruchi-pip Peri Pembo has fallen in the well. Like the first time, she can't hear him because the stream is too loud. He repeats himself, but she still can't hear, responding, "Tiresome Child, what are you trying to say?" He replies, at the top of his lungs, "Honorable Mother! Chari Bari Rembo Tikki Tikki Pip Pip has fallen into the well!" She says back, "Unfortunate Son, surely the evil spirits have bewitched your tongue. Speak your brother's name with reverence." Very slowly, out of breath, Chang says the name correctly. She says, "Oh, not my first and honored son, heir of all I possess!" She tells him to go get the Old Man with the Ladder. You would think he would have learned from his own experience in the well that his mother can't really help his brother get out, and probably should have skipped her and gone straight to the old man, who apparently cares less about propriety than about actually saving people's lives, but Chang is a young boy, so maybe it's beyond his understanding. But by the third time, she should have at least heard the part about falling in the well (she heard enough to know he was messing up his brother's name), or seen the emotion on her son's face and realized something dangerous was happening, so at that point she was just being horribly petty at the risk of her eldest son's life. Once Chang gets to the old man, he asks for help using his brother's full name, but finds the old man to be asleep. "Miserable child, you disturbed my dream. I had floated into a purple mist and found my youth again. There were glittering gateways and jeweled blossoms. If I close my eyes, perhaps I will again return." Chang finally just shortens it to, "Please, Old Man with the Ladder, please help my brother out of the cold well." The old man responds, "Your mother's 'precious pearl' has fallen into the well!" And runs as fast as he can. But by the time he gets the eldest out of the well, it takes TTTSRBRPP months to get better. "And from that day to this the Chinese have always thought it wise to give all their children little, short names instead of great long names."
From what I've read since reading this book, the Chinese do not have, and never have had, a tradition which dictates the length of their children's names. Nor do they wear kimonos or Japanese-style sandals. Unfortunately, many people see that this is supposed to be a Chinese folk tale, and assume it accurately portrays ancient China. Which it does not. Thankfully, the characters are not drawn as caricatures or anything that horrible.
However, even if this book wasn't culturally insensitive, even if it were set in a mythical land instead of a real one, it would still be a bad book. It depicts an institutionalized mistreatment of younger sons, suggesting they are worthless since they lack inheritance. If Chang was a girl (and thereby also ineligible to inherit property in many societies), this would be unbelievably sexist. As it is, there's no clear "-ism" for it, but I'm not reading it to my two sons. I both my children equally, regardless of their age, and throughout our current society, the vast majority of wills are written so that all heirs get the property divided equally among them. The mother belittles her younger son to the point of almost killing her favored eldest. She clearly plays favorites, and horribly neglects her children. It is a toxic message. It's supposed to be a funny story about long names, but I don't even understand what it's trying to say. What's the message the author's trying to get across? "Don't give your children long names"? That doesn't quite fit. "Here's why Chinese people have short names"? But that's not a thing. They don't have short names, any more than Europeans do. And it's not like it was a common misconception or stereotype, either. I'm guessing, based on her name alone, that Arlene Mosel is not Chinese. Although it's technically possible that she was, and married someone with the last name Mosel. Or heck, maybe "Mosel" is actually a Chinese name, and I'm woefully ignorant. But I'm disinclined to believe her as far as Chinese folk tales go, until she shows me she's done the research. And, according to my own sparse research, she didn't.
If the mother were punished for her callous treatment of her children, especially the youngest, then maybe her behavior would act as an anti-example. But as it is, with no repentance or personal consequences (only her eldest son suffers for it), I don't want this read to my kids. I don't want her behavior modeled for them. I don't want them to worry I'll ignore them if they fall down a well, or treat them differently based on their age alone. And I don't want them to feel that either of them is worthless. Not even in a relative sense.