Are application problems the only way to make math engaging?

Every time I say to a group of math teachers that I want to make math more engaging in my classroom…inevitably the first suggestion has been to incorporate more real world applications. Great idea but what about the content areas which don’t have grade appropriate application problems…like complex numbers.

Complex numbers or imaginary numbers are taught in sophomore year but I am yet to find an application problem…engineering? Not everyone finds it interesting…

This year I decided I am on a mission… I will find a way to make imaginary numbers interesting and engaging. 

I started the lesson by asking the students about the need for  natural numbers and why as humans we started using them. The students went on an exploration of invention/discovery of numbers, specially zero. Then we talked about the advancement of the human race and our evolving needs.
There came a point when we had all the numbers we needed but humans started applying those numbers  in geometrical shapes and constructing buildings. That is when we felt the need for irrational numbers. On the other side of the world, Al-Khwarizmi was writing the Kitab al-Jabr to teach the world about algebra. 
I had the most insightful conversation with my students about this. We discussed need of math for survival and need of math for exploration and discovery. That is when I introduced the concept of imaginary numbers as a way for humans to explore equations which do not have real solutions. 
This was the first time not even one student asked me why we learn this…their reflection at the end of the lesson was positive, showed engagement and understanding. Most importantly, this lesson marked an important milestone in our learning community as a turning point for the need and importance of mathematics. 
This is not to say that connecting content to real world application based problems is not the way to make it engaging, but I think when we focus ONLY on that, we miss out on the beauty of mathematics, its logic, its patterns, its creativity and exploration.

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