Slang is certainly teachable. The better question to ask is whether we should teach it at all. I think it’s a matter of acknowledging it and talking about it rather than teaching it.
In the digital world we live in today, text chats on phones and the Internet are very common. We all have seen the kind of shorthand writing that is a mix of letters, symbols, and emoticons. Some of us are more familiar with this SMS language than others. Our younger learners may even be more familiar with this “txt-speak” (text-speak) than any of us. IMHO (!), I think it is very useful to address the topic of Internet and text chat slang. We should make sure our students are very aware of how informal this style of writing is, how non-standard certain forms are, and how inappropriate it can be to use this slang in academic or professional exchanges.
In my ninth topic in the Language Notes series, my emphasis is on comprehension of slang rather than use. I strongly caution viewers about the dangers of lazy writing. One of my interactive exercises on my website asks students to replace all non-standard spellings with standard ones. Two others address common abbreviations and acronyms.