There are various types of questions that math students ask me and one could qualify them into various categories.. There are the almost rhetorical type questions such as 'what do we need to know this for?' Or 'Do you ever need this in your daily life?' These questions are often posed because students are struggling or are resistant to the practice required to understand and to continue moving forward in math. The next type of question deals specifically with the specific topic at hand. The third type seeks to understand the underpinnings of math: why it works, how it came about and so on. I had a question of this type the other day. A student was asking where the concept of degrees came from and we discussed it's possible roots in that there are approximately 360 days in a year and that the earth moves approximately one degree each day around the sun. A fourth type actually aims at advancing math itself! I've had some of these questions, where students actually aim to come up with new mathematical formulas and they run them by me.
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