Charcoal represents a realistic substance, which is our common black solid. Charcoal is mainly composed of carbon, but it is not entirely composed of carbon. There are other impurities such as sulfur in charcoal, which is a mixture.

Carbon stands for a chemical element with the symbol c. Carbon represents a pure substance, a simple substance, which can react with oxygen to produce CO or CO2.

Strictly speaking, the main component of charcoal is carbon, so charcoal+oxygen is equivalent to carbon+oxygen.

Strictly speaking, charcoal is a mixture. Charcoal+oxygen should produce many substances, not only CO and CO2, but also charcoal can't be written as C without mixture in chemical equation.

So strictly speaking, your teacher is wrong.