Learn French: Nous prenons conscience du problème. - We are becoming aware of the problem.

The French expression prendre conscience de translates literally to "to take awareness of". In English, it sounds more natural to say "to become aware of" or "to realize". French also uses the simple present tense (prenons) for ongoing actions where English uses the present continuous ("are becoming").

Why isn't there an article before conscience?

Prendre conscience is a fixed expression. French often drops the article in verb-noun combinations with prendre, just like in prendre soin (to take care) or prendre peur (to get scared).

Why use du instead of just de here?

The phrase requires the preposition de (prendre conscience de). Because you are becoming aware of a specific masculine noun (le problème), the preposition de and the definite article le must contract into du.

How would we use this lesson's "same subject" rule to say "We realize that we have a problem"?

You might intuitively try to say Nous prenons conscience que nous avons un problème. However, because the subject (nous) is the same for both actions, French prefers replacing que + clause with de + infinitive: Nous prenons conscience d'avoir un problème.