Curriculum Development to Promote Visualization and Mathematical Reasoning

What first comes to mind? For you is 2 a number? Is it a radical? Is it a measure? Is it the length of the diagonal in a unit square? Is it the length of the hypotenuse of a unit-length right triangle? Maybe it is the secant of 45o? How you imagine 2 is dependent on the experiences you have had with it. The more limited our experiences with a mathematical object the more limited our understanding of it. The broader our experiences with the object the greater our understanding of it. 2 is an interesting object. What is it? What does it look like? How does it behave? What might a mathematics teacher do to provide opportunities for learner activity that broadens the learner’s relationship with 2, making possible greater understanding of it and greater understanding in general of radicals?