Villager Farmers will collect, plant and harvest Wheat and Beetroot, but they will not harvest Potatoes and Carrots when they're fully grown.

What I expected to happen:
If you give a villager farmer wheat seeds, he will pick up plant those seeds in nearby farmland. When the wheat is fully grown, he will harvest the wheat and replant the seeds.

What actually happened:
If you give the villager carrots or potatoes, he will pick them up and plant them in farmland, but he will not harvest them when they are fully grown.

Steps to Reproduce:

• Place villager farmer in a pen with tilled soil
• Throw stacks of Carrots or Potatoes to him
• Observe the villager planting them
• When the crops are fully grown, notice that the villager doesn't harvest them as he should

^{Reference: http://minecraft.gamepedia.com/Villager#Farming}