You did not mention the age of your student. I will assume he is around ten.
Most children at this age will have great resistance to reading music, because they have not reached the stage of fluent pattern recognition and interval reading, They are laboriously deciphering note by note on the score and probably they do it only during the lesson, which typically happens once a week.
I certainly don not teach by the Russian method (actually I do not even believe in the existence of a Russian method!), but I do start teaching by rote because students want to play, and if we are going to wait until they can read music, they will give up!
However, I believe that reading music is the single most important skill a student can acquire so we are at it from the very first lesson. Even if it is a small child who cannot read and write, I will teach music reading (Why should reading language precede music reading? Music reading is far simpler). I think Anda mentioned in one of her posts that she teaches note recognition on the score before note recognition on the keyboard. I completely agree and I follow the same approach (I think): First they are taught to recognise the names of the notes on the score. At the same time they are taught to recognise the names of the keys on the piano. Only when can they do both well do I start to show the correspondence between score and keyboard. Once they can do that we start interval recognition and reading by following the “curves” on the score rather than by reading individual notes. Other things are going on at the same time, most importantly recognition of scales and chords. It all follows a very systematic and organised procedure and by the end of anything between three months and two years the student should be able to sight-read music more or less fluently, but certainly without difficulty. All the while we continue working on pieces, and if need be I will teach by rote, while I wait for the reading to catch up with the playing.
Now for your question. It seems to me that the Russian teacher may have had a similar approach, and was working on reading as well as playing, but by the time the student got to you the reading had not caught up with the playing (don’t pay too much attention to what the parents say – their description is not informed enough). After all she was with him for only six months.
Memorising is good. It is not a question of reading X memory, but rather now that he can memorise well he should bring his reading skill to the same level.
Best wishes,
Bernhard.