[Rutherford]

Could someone rationalize the first step in this synthesis? What is happening there?

[discodermolide]

The first step is a simple dialkylation of the malonate ester with methyl iodide.

[Rutherford]

How does it work?

[discodermolide]

The base, NaOEt removes one of the acidic protons of the CH₂ group. They are very acidic due to the electron withdrawing esters. The resulting anion is stabilised by resonance through the ester carbonyls.
Methyl iodide is the electrophile and is attacked by the carbanion to give the methylated compound.
Rinse and repeat for the remaining CH to give the dialkylated compound.

[Rutherford]

And the last step is the same process then. Thanks.

[discodermolide]

No problem.