Can our brain be trained?

You can negatively train your brain with negative thought patterns such as:
I'm not good enough
I'm not a good test taker
I'm not a good friend
Then these messages you tell yourself become your self fulfilling prophecy and you have trained your brain to believe them
You can train your brain to reaction to stimulation. For example Pavlov's dog who salivated at a bell because the brain had been trained to react to a bell because dinner was soon

I image you are meaning more training your brain to accomplish things though and in that sense science says you can.
You can train your brain to memorize and recall massive amounts of information rapidly. Some memory world records are:

• 1. That is how many digits Wang Feng (China) memorized in just 60 minutes. They were consecutive digits
• Simon Reinhard memorized a deck of 52 shuffled cards in just 20.44 seconds
  These world records were achieved by brain training and most champions use the Mind Palace

Athletes train their brains all the time with visualization. They will sit back and image hitting the baseball, shooting baskets, boxing, golfing or whatever their sport is. They are training their brain to visualize success and get used to seeing the success. Studies have been done that suggest visualizing can be equally as effected as the actual physical training and doing both visualization together with physical training has better results than only doing one of the items Seeing Is Believing: The Power of Visualization
Speed reading is an example of training the brain. The average brain will read 250 words per minute. However, when you do simple things such as forcing the brain to move faster than you think possible the average person can double or even triple their reading speed

In conclusion, your self talk trains your brain. You can train your brain through expected stimulation and even train your brain to memorize or speed read. Visualization is a powerful way to train your brain to expect and model success

These are just a few ways to train your brain