Hermione had sworn to Dumbledore that if Voldemort ever did find out about her heritage, she would NEVER join him. Nevertheless, Dumbledore felt it better to train her for the when Voldemort DID find out. To help her be more prepared. So he enlisted Remus to assist her.
Her lessons were usually the development of her new powers, such as telepathy, which enabled her to move items with her mind and elemental power, meaning she could actually control the elements. Elemental power was harder to control. For now, she could only control wind. Wind was the easiest to master. If she just closed her eyes and concentrated, she was able to cause a small breeze and then it would grow stronger and stronger.
In the beginning, Hermione was slightly resistant and awkward when training with Remus. Since she was Voldemort's heir, Dumbledore explained that her body and powers would go through a few changes when she was of age. So after her sixteenth birthday (celebrated with Tonks, Remus, McGonagall, Dumbledore and Moody at Grimmauld Place), Hermione had noticed that at times, she would feel as if an electrical charge was running up her spine. Dumbledore called it a power surge.
Her hair had also flattened out and grown darker and darker, until it was the color of ebony. Her cinnamon colored eyes, which Dumbledore said she had inherited from her mother, remained though. Her warm eyes and dark hair looked odd together but in a way, they suited her well. Her skin had also grown slightly paler and her lips took on a fuller, pouty blood red style.
The first few days, Hermione was completely embarrassed about her new look and for a few comical hours (at least it was for Remus) actually wore a brown paper bag over her head to hide her changes. Eventually, Hermione grew to accept her looks like her heritage. Her powers were a bit harder to control.
At times, when Hermione felt angry, she felt that she did things that she couldn't control. Like when Remus was forcing her to have extra late night training and she was so furious that Remus wouldn't let her sleep that she subconsciously caused his sleeve to light on fire. That was the first time she ever controlled fire. She apologized profusely but Remus was more proud than angry: fire was a very hard element to control, almost the hardest (lightning was the hardest).
Later on, Professor Moody had suggested that Hermione learn advanced wand movement and dueling techniques so even more work was added to her schedule. And that was how it went: development of her powers in the morning with Remus, dueling in the afternoon and at night she could have time alone to relax or study. And that finally brings us to the scene with the sponge.