Nonetheless, you can read the author's original argument here, where he argues flashcards are useless as they 1. create a particularly slow set of associations in the learner's brain inappropriate for actual language use and 2. wrongly reinforce the notion that there is a one to one correspondence between words in different languages. My original comment follows.
Both counts are wrong.
1. The Slowness of the Association
I'm not sure how other people experience it, but when I learn a language it's only at first that my brain goes through the long slog of conjugation charts, etc. For example, let's say a flashcard contains the word "dūcet."
Now when I'm first learning the word, I'll think:
1. dūcere means to lead
2. third conjugation, so the -e infix represents the future tense.
3. -t ending gives third person singular.
4. Now (in my head) I reach the English phrase "he will lead."
5. (then my brain will very quickly decode what is meant by "he will lead" in English. Since I grew up speaking English, this is fast enough to be automatic. I may see a mental picture of a man leading, or just have a wordless sense of what it means.)
This is a slow process. But as the flashcard comes up again and again, the different steps get elided over. Each step becomes as automatic as the last one, the one where an English sentence is decoded. Until, within a short amount of time, when I see dūcet, I simply understand it means (he will lead). Or when I see a -t ending in general I think third person. And really this is all we do when we learn languages--we gather information, learn the patterns from that information and apply them.
Really I don't see how this is any different than any other method of learning vocabulary. When I was a child my parents would point to my ears and ask what they were. At first I would have to think about it, just as I had to think about dūcet, but eventually the association became fast enough to be automatic. And now I just think "ears." If I'm in the middle of a Spanish sentence, I think "orejas." Associations between concepts and words have to be formed--it's simply a question of the best way.
The benefit to flashcards, especially the Anki program, is this process is very quick, not to mention the added benefit of it scheduling easy words rarely and hard words often. I don't need to go find my parents and have them point at my body parts. I just click a button, and it takes half a second.
Moreover, I can create a massive amount of flash cards and learn a massive amount of vocabulary in a single day with Anki--something that would be much slower if I had to learn through some alternative way--looking at pictures, or reading Wikipedia articles on the topic, etc. I can't conceive of a faster way.
II. You'll Believe in Word to Word Associations.
Well, yeah, if you believe the cards represent that, then you'll come to believe that one word is a perfect substitute for another. But why would you believe that in the first place, since it's, as you point out, so patently false?
Let's go back to dūcere. On the back side of the card, it says "to lead." Does that mean that I believe "to lead" and "dūcere" are exact translations? Of course not. I realize that the words mean roughly the same thing, but that of course the overlap is not perfect. Perhaps dūcere means several things (it does)--in which case, I simply add other translations to the back of the card. Or perhaps it means one thing in one context, and one thing in another? In that case one card can be "dūcere [referring to time]" and "dūcere [referring to people]."
Again, the only reason that a belief in a one to one correspondence between the term and its translation would get reinforced is if you already believed that in the first place. If instead you're intelligent enough to realize all translations are necessarily somewhat inaccurate (See Quine), then there's no problem. I know that the answers are just rough translations (though I can make them very specific if I like).
Plus you're somewhat caricaturing the flashcard system with the example of three as "три" or "трёх". That's not a problem: it's easy enough to write "three (gen)."
Now the answer is only трёх. The reverse card would be трёх with the answer "three (gen, acc an, prep)" [genitive, accusative animate, prepositional]. You can make your flashcards as precise as you like. One of my Latin cards, for example, reads "large sea [animal], anything ranging from a shark to a whale". Cēta, ae, f.
It's true that this definition will be missing something. But so will any association [got] through any method! If you only learn your words by, e.g., reading articles about that topic, then you will have learned the meaning of the word in that context--but you don't know every context in which the word will show up, either, and so the association you've formed is partial as well.
Dictionaries, however, are specifically designed to collect the meanings of words (in all their contexts). Thus, paired with flashcards, they serve as very direct ways of learning the usage of words, as opposed to other systems not specifically designed to capture and teach the meaning of word. I could, to the contrary, listen to a word in several sentences and take a guess at what it means--but it's much easier to just open a dictionary or ask an English speaker.
In short, flashcards are a great way to learn. They are not sufficient to master a language--if you want to be better at writing, you'll have to practice writing, if you want to be a better speaker, you'll have to practice speaking--but are a very useful tool.
The author replies in the same post linked to above.
Update: I went back to the original post to respond to the criticism of my original comment, but it appears that comment has been deleted (though it still shows up on a Google search). I don't know why, but I suppose it means I was right about something.
My comments at the site in question are now being altered by the site author. For the record, the other comment I left read, before being edited:
It is, of course, your blog, and you're welcome to enforce whatever policy you like, but I think it's a shame that debate on such an interesting and important language issue was shut down.
I've also preserved my comment (I only made one) on my own blog: http://scheule.blogspot.com/2010/11/on-flashcards.html
Readers are welcome to judge for themselves whether the comment was useful or merely argumentative.