Being around mathematics all the time, one develops a sense of mathematical awareness...what mathematics is, what it means to do mathematics, how mathematics is used, etc. In one sense, we become spoiled and perhaps set in our ways.
Bret Victor is an individual who can give you a jolt...force you to rethink your assumptions, your actions, and your complacency. Be forewarned...his ideas may not fit with your current world.
First some background. Educated as a electrical engineer (U of CA-Berkeley and CalTech), Victor became a "design engineer." Choosing two examples of his work from many, he designed the initial interface concepts for the iPad and won the Apple Design Award for his interactive data graphics for Al Gore's war on climate change.
Victor's overall website is a creative wonder to behold and explore, but I am going to suggest you start with his Kill Math efforts. In this project, Victor is pursuing "techniques that enable people to model and solve meaningful problems of quantity using concrete representations and intuition-guided exploration. In the long term, I hope to develop a widely-usable, insight-generating alternative to symbolic math." Thus, the term "Kill Math."
Read and reflect on his message. Be sure to visit the links he suggests as examples:
I must agree with Victor's concluding remark: "Mathematics, as currently practiced, is a command line. We need a better interface."
- His Scrubbing Calculator
- His interactive demostration of the differential equations underlying the predator/prey relationship
- His essay "Up and Down the Ladder of Abstraction"
- And my favorite is his simulation (and extension) as a replacement solution of a common math problem
- His collection of "Things Other People Have Said" also is refreshing
Any teacher who has watched students suffer trying to use the awkward symbolism of mathematics (e.g. 2+(-2) and f-1(x)), will probably agree. Victor offers a lot of food for thought...and some of his thoughts will disturb you (perhaps rightly so!).
And, once you have recovered, you might explore Victor's full website...it is a visual mental trip.