TPRS is based on comprehensible input, and believes that students learn by input.
Other major approaches to date have been based on output, and believe that student learn by outputting language.

So, the two are not really compatible. They are based on completely opposite things. It’s true that many factors come together to support acquisition, but in the main, there is a single mechanism that makes people fluent in languages. And since everyone learns the first language without exercises, pairwork and forced output, my vote is on input.

If you believe one mechanism is what makes people better in language, you will naturally want/need to devote the bulk of your teaching time to that approach, because that’s the one you believe will lead students to acquire language.

“Mixed” teaching is not as effective as TPRS in having students acquire STRUCTURE (aka “grammar”) as quickly as pure TPRS. This is because TPRS is a tool specifically designed to put structure into heads as rapidly and PERMANENTLY as possible. It is not a tool to stuff a lot of vocabulary into heads at the price of losing fluency. Students can use the language we have given them through TPRS automatically and correctly. They CANNOT use the language they get through other means automatically and correctly — watch those eyes looking upward as the student “searches” for the word, often getting the pronunciation wrong because the sound is not firmly in the mind.

“Mixed” teaching puts more vocabulary into heads more quickly than pure TPRS. And that looks great on tests and on curriculum documents. Look how rigorous this course is! But that isn’t acquisition. It’s learning — memorization. The question is — how PERMANENTLY are those words in place for ALL the students? The “star” students will be able to hold onto words they learn with less repetition, if they are interested in them, but we are teaching all the students, not just the star students. Even the star students will lose a lot of the words they memorize for class and tests, unless they are reinforced with serious repetition — which most non-TPRS or non-CI-based approaches do not value.

TPRS produces students who are “micro-fluent” — they can use whatever subset of the language has been done through TPRS unconsciously and correctly. Other approaches do not produce this kind of micro-fluency. It produces students who know things. How long they know them is another question. If they happen to get enough repetition, they will know things for a long time — but in most former approaches, there is so much emphasis on “more vocabulary” that no one vocabulary word gets the repetition it deserves and needs to be acquired, not just memorized for a time.

Micro-fluency is an amazing tool. It allows students to use native-like strategies in their reading, rather than decoding word by painful word (this is particularly true in Chinese, where there’s an extra layer between the written form and the meaning because of the non-phonetic nature of the script). It means that student errors are “logical” errors in speech and writing. It means no more “me llamo es”.

For Chinese, the ideal program based on my experience would be two solid years of CI/TPRS, then hand over to a native-speaking teacher who can make those micro-fluent kids really, really literate. They will be reading easily and fluently the language they have acquired when they make “the switch”; but with that foundation in place, and confident in their reading and language in general, they will be able to easily and quickly expand their vocabulary to the levels that are needed to face some of the “big tests” (AP/IB etc).

But, just as a child is fluent in English before going to school to learn academic English, we must give them that fluency before we start in with the details and before we expect a lot of words.

Krashen’s i+1 is frequently quoted as support for giving unknowns. But the “+1” part is always going to be STRUCTURE until the student has internalized all the structure of the language. If that student isn’t fluent in the grammar of the language, without thought, then you have very little scope to add anything new because the structure is always the most basic part of the language, and will always take up that “+1” position. Knock the structure out of the way, wrapping it in the highest-frequency words of the language, and you will then be free to open up the “+1” to lots and lots of new vocabulary, collocations, and literacy elements.

The problem with most non-CI-based approaches that are adopting the ideas of “comprehensible” and “communicative” is that they can’t bear to let “+1” be “+1” and not “+100”.