Stephen Krashen himself has joked in a self-deprecating way that he came up with the vague concept of “i+1” to achieve fame as people argue its meaning indefinitely. Before contacting him, I wasn’t exactly sure how seriously we should take the man! Thankfully, Stephen clarified that for me real quick. Regardless of the joke being aimed at those using academic jibberish, the concept of “i+1” is demystified in the following Krashen-approved commentary…
Many (most?) people referring to “i+1” incorrectly state that “i” is what a learner understands, and “+1” is the next bit of language a learner doesn’t understand. This is not at all what Krashen meant. I’ll admit that I absorbed what those people had been saying over the years, even joking with Latin teachers about how ancient unadapted Latin can feel like “i+10” (except for Tacitus, which feels like “i+100,” right, Johns?). Anyway, I have no problem admitting being totally wrong on that. It just took some time reading the primary source, and cross-referencing findings and ideas with other research. The result? Well…
If a learner doesn’t understand the input, it doesn’t even qualify as “i+1” yet.
That’s right.
The learner must understand language that contains structure just beyond the level of competence. This doesn’t mean that the language should be difficult to understand at all! This also doesn’t mean that learners should only be receiving that kind of input (i.e. input that contains structure just beyond the level of competence). So, an entire class with no new language that’s completely understandable is fine (e.g. instead of 3 new phrases every class, like clockwork, in an attempt to cover a department’s thematically themed vocab and grammar-based curriculum, etc.). Support for this can be seen in research done on Extensive Reading, of which a core principle, at least according to Jeon & Day (2015), is “books are easy.” If books are easy, learners are unlikely to find much “i+1,” but that’s not even an issue according to Krashen’s hypothesis; learners shouldn’t only be receiving that kind of input, because it will be provided automatically over time. The pedagogical implications are that it would be easy for teachers to adopt a content-area mindset of “challenging students” by giving them something hard to read. The problem with this mindset, however, is that none of our research suggests it’s a good idea to do so!
So, CI is input that’s understood, and its relation to acquisition is “i+1.” Those are not the same thing, and we should avoid using the concept to justify giving learners reading that’s above an appropriate level.
Here are Krashen’s precise words on that:
We may thus state parts (1) and (2) of the input hypothesis as follows:
(1) The input hypothesis relates to acquisition, not learning.
(2) We acquire by understanding language that contains structure a bit beyond our current level of competence (i + 1). This is done with the help of context or extra-linguistic information.A third part of the input hypothesis says that input must contain i + 1 to be useful for language acquisition, but it need not contain only i + 1. It says that if the acquirer understands the input, and there is enough of it, i + 1 will automatically be provided. In other words, if communication is successful, i + 1 is provided. As we will discuss later, this implies that the best input should not even attempt to deliberately aim at i + 1.
Magister P. 
So, let’s use an example from English language acquisition. A toddler hears her father say, “Do you want a cookie?” This input is comprehensible as the word “cookie,” combined with the interrogative intonation of the father’s voice, previously has resulted in the child’s delight in receiving a cookie and the ensuing sugar high. The “Do you want . . . ” element of the communication is input beyond the toddler’s actual comprehension. Does this entire question, therefore, constitute “i+1”? If the toddler mimics the communication while holding a cookie with the expectation that *she* would receive a cookie when, in fact, her father says “yes” and eats the cookie himself thus demonstrates the “i+1” sophistication of such a phrase. Through repeated use over time, the toddler learns that “Do *you* want . . . ?” element results in different outcomes depending on the one who asks.
What do you think he means by competence? Is it competence in comprehension, like Randy suggests above? Or competence in output.
If it’s competence in output, then almost any grammatically correct utterance would be i+1, right? Until you get to probably intermediate high, I guess.
See Vygotski’s Zone of proximal development: https://en.wikipedia.org/wiki/Zone_of_proximal_development
It’s always get to hear your thoughts about comprehensible input. I think it’s really valuable that you’re part of this dialogue around applying CI to classrooms.
I think there may be a misunderstanding among Latin CI teachers about how much vs how little to shelter. The Krashen example quoted which says that input doesn’t have to only contain i+1 is intended more as a criticism against sheltered grammar curriculums than as a recommendation to only include a mixture of only i+0 and i+1. We can see this in his section about ‘caretaker language’ on pp22-23 of the paper you linked to (http://www.sdkrashen.com/content/books/principles_and_practice.pdf) where he makes it clear he is criticising the over-sheltering of a grammar curriculum:
“An interpretation of this finding is that caretakers are not taking aim exactly at i + 1. The input they provide for children includes i + 1, but also includes many structures that have already been acquired, plus some that have not (i + 2, i + 3, etc.) and that the child may not be ready for yet. In other words, caretakers do not provide a grammatically based syllabus!” (Krashen)
The point is that an utterance can be comprehensible in broad brush-strokes and yet contain many features that are unknown and not even the next step they are ready to learn. Krashen makes a point that caretaker language is only ‘”roughly-tuned” to the child’s current level
of linguistic competence, not “finely-tuned”‘. In which case, if we are mimicking natural language acquisition circumstances, we should aim to keep the overall comprehension accessible but not be worried if not every word can be understood, because in natural circumstances caretakers do not achieve comprehension of every single word with their child and limit their input to just i+0 and i+1.
I agree, and the pedagogical implications are that we have NOTHING close to those natural conditions with just an hour or so each day (if we’re lucky). And so if kids are just kinda sorta understanding mostly just the gist, we’re not curating the input enough as teachers.