The quote actually comes from Mivchar HaP'ninim, which was written originally in Arabic, then translated into Hebrew by Judah ibn Tibbon (father of the famous translator of Maimonides' "Guide", Samuel ibn Tibbon). Perhaps one reason why a quote such as the one you saw became popular was due to the publication of an English version of the book (available here) in 1859 under the title "A Choice of Pearls", translated by B. H. Ascher.
The first chapter of the book is concerning the nature of wisdom, written in the style of questions asked to a wise man and his pearls of wisdom offered in return. One of the the wise man's sayings is:
.ואמר תחלת החכמה שתיקה. שניה לה שמיעה
.שלישית זכרון. רביעית מעשה. חמישית מורה
Ascher translates this as:
The first step to wisdom is silence; the second
attention; the third memory ; the fourth activity; the
fifth study.
This translation is similar to the quote you have above, except for the fifth step, in which Ascher translates "מורה" as "study", which is not how I would translate it, which would be more in line with your version: "teaching". [I have no access to the original Arabic, nor would I understand it even if I did, so I'm trusting Ibn Tibbon's translation.] It's odd, because in the Hebrew footnotes he compares with manuscripts that have "למורה" as the last word, which he clearly understands to mean "to teach others". Also, he cites one manuscript that has "מודה" (~"acknowledgment") instead of "מורה", in which case the fifth step would be something like "acknowledge the truth".
