There are excellent reasons for your confusion! In fact, your teacher is correct.
Blood changes colour because different levels of oxygen cause the haemoglobin molecules to change shape. Different molecular shapes alter the way that light is absorbed by the haemoglobin and thus the colour of the blood. Deoxygenated blood is a deep red colour, while oxygenated blood is a brighter, more orangey red. To make a distinction, medical illustrations often depict the deoxygenated blood, which travels in the veins, as blue.
Moreover, when a white skinned person looks at his or her veins, say on the wrist, the veins appear to be blue. Toronto physicist Dr. Lothar Lilge and his colleagues at the Ontario Laser and Light Wave Research Centre set out to find out why. They created fake vessels by filling glass tubes of varying sizes with blood. They then immersed the tubes in a milk-like fatty fluid. The fluid was chemically similar to fair-coloured skin in the way it reflected light. They found that the artificial vessels appeared to be red when they were on the top of the milky fluid. But as the researchers lowered them slowly into the liquid they appeared slightly bluish. A scholarly paper by Lilge has more information about his experiment and this phenomenon.