The characteristics of mathematical communication: Rigor-ousness, Economy, or Freedom which they participated in communication by deictic, iconic, beat, or metaphoric gestures.
The characteristics of mathematical communication includ-ing Rigorousness which referred to one’s opinion expression, speaking and talking, and discussing for sharing one’s mathe-matical ideas expressing step by step in mathematical problem solving, and being able to send and receive message congru-ently with one’s idea. Economy referred to one’s opinion ex-pression for sharing the mathematical ideas concisely to the others in mathematical problem solving, and being able to send and receive the concise message as well as make the communi-cation participants have common understanding. Freedom re-ferred to one’s opinion expression, speaking and talking, and discussing for sharing various or new mathematical ideas in mathematical problem solving.