Meaning and Underlying idea of a definition or a theorem

Question

What does it mean by 'explain the meaning and underlying idea of a definition or a theorem'?

For example, if we are asked to explain the Fundamental Theorem of Algebra, how should we explain its meaning and idea?

I can only think of saying: The theorem tells us that every single variable polynomial can be expressed into a product of linear factors.

What else should I say? I think I am just stating the theorem, nothing more. I am very confused right now.

Answer 1

To answer your question I want to cite this answer to a question on matheducators.stackexchange.com:

"Math educators mostly agree that for a given object or property, people usually have a picture in mind (maybe many pictures). We call this their concept image. The students also might have a concept definition in mind, hopefully the definition you gave to them." – Anschewski

The formal definition or theorem stated in the textbook is the concept definition. You are asked to give also the concept image of the theorem. The following questions might help you:

• What do you have in mind, when you think about the theorem / definition?
• Why is this theorem important?
• For what is the definition / theorem used for?
• What is the motivation behind the definition / theorem?
• How can you illustrate the definition / theorem?

The answer by Anschewski also links to the book Ideas from Mathematics Education where you might find some help.