The word mindfulness (sati) does not mean contemplating (anupassana). 

Mindfulness refers to 'remembering' to keep the mind clear of unwholesome states. 

When mindfulness keeps the mind clear of unwholesome states, the act of 'contemplating', i.e., 'seeing closely', naturally occurs. 

Then when body, feelings, mind & Dhammas are 'seen closely', mindfulness again acts to ensure or 'remember' to abandon any wrong views that may arise towards body, feelings, mind & Dhammas. 

Also, contemplating (anupassana) is a state of 'seeing', i.e., a state of 'non-active-thinking'. When the mind only sees & does not actively think, the mind calms down, the breathing calms down & the body calms down. 

Thus, when when mindfulness *maintains* clarity & contemplating, certain 'nirodha' ('cessation') processes naturally, occur. The breathing becomes calmer & calmer, longer & longer, more & more refined. Internal stored stress progressively comes into awareness & dissolves. This calming & purification build a certain momentum, resulting in mindfulness being easier & easier to maintain because the mind, breath & body are becoming calmer & calmer; purer & purer. The happier & more content the mind is, the less effort required to maintain mindfulness. 

When the body becomes very purified from stored mental stresses & formations, jhana will occur. This can be compared to continuously using mindfulness to dig out dirt. When the dirt is dug out,  water naturally flows into the cleared space. This naturally 'flowing' of the mind's momentum into a unity is jhana.  

In summary, mindfulness acts to remember, moment by moment, to keep the mind clear of unwholesome states & wrong views. This constant maintenance of mental clarity then, eventually, becomes jhana.