How to work with enharmonic training
The module shows a note and asks for its enharmonic equivalent, that is, the other name for the same pitch. The first levels deal with simple pairs such as C♯ and D♭, later ones with double accidentals and with the question of which spelling is correct in a given context.
The exercise is short and works well as a warm-up before larger sessions. Five minutes is enough. More important than quantity is that you briefly justify each answer: why one spelling fits a key and the other does not.
If you have an instrument to hand, play the note along. That makes it immediately tangible that both names land on the same key or the same fret, and that apparent contradiction is the heart of the topic.
Why one note has two names
In equal temperament, which has governed practice since the eighteenth century, the octave is divided into twelve exactly equal semitones. As a result, notes that are theoretically different coincide: C♯ and D♭ sound identical, and so do B♯ and C. That is called enharmonic equivalence.
Theoretically these notes are not the same thing. In equal temperament both are the same pitch, but in the older tunings they are not: in Pythagorean tuning C♯ lies about 23.5 cents higher than D♭, which is the Pythagorean comma. In meantone and in just tuning C♯ lies about 41 cents lower than D♭ instead, which is the greater diesis, nearly a quarter tone. Historical instruments with separate keys for C♯ and D♭ really did exist. Equal temperament is a compromise that makes every key playable at the price of detuning each one slightly.
In practice, context decides the name. A note that wants to resolve upward is written as a raised note; one that pulls downward as a lowered one. That is why a cadence into D major uses C♯ and not D♭, even though both sound the same. Notation describes not only pitch but intention.
From practice
Enharmonics get dismissed as hair-splitting until the first chord symbol stops making sense. When a lead sheet says D♭ major and the student plays C♯ major, everything sounds right, but suddenly the next line no longer matches the notation. At that point the purpose of the distinction becomes clear.
A good way in is enharmonic reinterpretation as a means of modulation. A diminished seventh chord can be read four different ways and leads into a different key depending on the reading. That principle carries a large part of Romantic harmony and becomes comprehensible as soon as you take the names seriously.
For daily practice one rule of thumb is enough: in sharp keys write sharps, in flat keys write flats. Follow it and you will be right in ninety per cent of cases, and the remaining ten per cent are precisely the interesting ones.
Common mistakes
The most frequent mistake is treating enharmonics as interchangeable. The two names are not equivalent; they are right or wrong depending on context. On higher levels the module asks about exactly that.
The second mistake is avoiding double accidentals. The equivalent of F♯ is G♭, but the equivalent of F𝄪 is G, and of C𝄫 it is B♭. Such cases look contrived but follow necessarily from the system and do appear in real scores.
The third mistake is assuming enharmonics are a property of the piano. They are a property of equal temperament. On a violin or in a choir, C♯ and D♭ really are intoned differently, and experienced ensembles do so audibly.