To avoid these run-ins against shelves, trash bins, pots, etc., you'll have to play a mini-game that involves solving a math operation very quickly. In the easier levels, a couple of sums will do the job but as things get more and more complicated, you'll have to do several multiplications and divisions.
The High Road's storyline is very important, letting you discover what happened in the building where the cleaning lady works. The dialogues between the protagonist and the rest of the employees in the building are the perfect link between the plot and the gameplay.
The High Road is a totally different experience from what most players are used to seeing. Its story, protagonist, and gameplay are a breath of fresh air for the RPG genre.