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I was going to be tutoring "boat people." Largely illiterate, from poor rural areas.
So a speaker of some language went up in front of the classroom with a few objects. Simple, right? Point and say the word. So she pointed and said a word. She had us repeat it. Then she taught us a couple more words.
We were asked what the words meant. Were they the name of the object, their colors, materials, how she was holding them, their size or some other attribute (big, smalll, old, new, important, cheap...), their uses?
Point taken.
Years later I saw one of Kenneth Pike's "monolingual demonstrations." He was a linguist, and a damned good field researcher. The linguistics dept. had brought in a black woman who was pretending to not speak English. Pike had 60 or 90 minutes to prise out as many words and as much grammar as was possible. He had no idea what language she spoke.
He started everywhere: He'd hold up an object and transcribe what she said. He'd manipulate the object and transcribe. He'd alter the object and transcribe. He'd carefully selected his objects: Some were almost the same, with just a few things different--size, color, material. In 30 minutes he had a fairly good idea of the phonemic inventory, had worked out a few grammatical morphemes, and had several meanings proposed for a number of words. He then proceeded to eliminate possible meanings while teasing out more words. Her skin color and the language she spoke confused him. She was obviously African in ancestry; the language was utterly un-African. He worked out a few dozen words, had a few dozen more with suggested meanings, a reasonable sized chunk of the grammar, and his phonemic inventory wasn't half bad in the time he had. Never did guess the language: Garifuna. The woman went through and "graded" his work--he got one word wrong, but otherwise was good for his "I'm pretty sure about these" words.
Pike's approach is how you'd have to do it. Start with concrete objects carefully chosen and see what they say. Form hypotheses and test them. As you go, sort out what attributes are being taken as most salient; as you discover those, you learn things about the culture (and in the case of aliens, about their sensory apparatus and what they think is most perceptually salient). Keep it simple, work out the grammar and what's important before you try to build a hefty lexicon.
Math is a good basis for some communication. Mostly it's there to demonstrate sentience and level of intellectual sophistication. Any race that's made it to space will have worked out a lot of math; mathematical relations are presumably either universal or universally decipherable, once you've worked out the symbols. Won't help with, "Hi, I'm Bob, are those large lumps on your chest really breasts?" or "You really should have warned me that the empanada was full of boiled grub paste, I only like roasted grubs."
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