This leads to a much more diluted Martini. Dilution has an effect of enhancing the vermouth, so in a shaken Martini, add even less vermouth than you normally do. With very juniper-forward gins, particularly unfiltered ones, it will also "louche" that makes the Martini slightly opaque like a pastis – this is perfectly normal. If shaking, make sure you only shake for about 15 seconds (until the shaker is ice cold to touch) and don’t shake too hard as this will shatter the ice and cause even more dilution from the shards of ice. When you serve it, you should ‘double strain’ it which means holding back the ice with the Hawthorne strainer and pouring it through a fine strainer (a small sieve) to catch any of those inevitable ice shards.
This is generally the preferred method, as it allows some dilution but not too much. You need to stir for around 30 seconds. It all depends on ambient temperature though, so make sure your Martini is cold and stir more or less as required.
Or the Duke's Bar method. Freeze the bottle of gin. Rinse the glass with vermouth (add 1-2 bar spoons of vermouth and coat the entirety of the inside of the glass before discarding excess – if the glass is properly frozen then the vermouth should stick to it leaving nothing to discard (this is how they stop carpets getting sticky!), then simply pour frozen gin in. This leads to a very dry and very cold Martini but no dilution.