A common mis-conception about the field of mathematics, especially among students growing up, is that the field has reached its limits and there nothing more to be discovered or no problems that have gone unsolved. Math is a poured set of concrete that has settled, never to be changed over time.
It is this sort of mis-conception that allows students to enter into a math course without any excitement or anticipation. Any one with a somewhat advanced interest in math though knows that this is not the case and that mathematics is a field paralleling the sciences in terms of modern breakthroughs and ancient problems yet to be solved.
At the beginning of a course a teacher has the perfect opportunity to convey this sort of thinking to their students in order to spark their interest and let them know that this course is going to be engaging. A great way to do this is through the use of multimedia which often is over looked in the math classroom. If you can set the tone early as a teacher, then you have the opportunity to create an engaging classroom in the future.
A great concept to do this with is the concept of infinity. Most people have heard of the concept but have not taken the time to try and wrap their minds around it. It is when you do that things get interesting. Take the following video for example:
The video is an exploration on how big infinity is. This is a perfect chance to begin a discussion with students: what does infinity mean? How big is infinity? The video explains how there are infinities within infinities which means that there are larger and smaller infinities. Surely this will get some of them interested. The video then goes on to explain how "there are unanswerable questions in mathematics". Suddenly the solid cement has melted and mathematics is now a field much like the sciences where inquiry is encouraged and questions remain unsolved. This short discussion and video can prove fruitful for future inquiry based activities in the classroom.