Let R be a (nice) region in the plane with boundary curve C, and let P(x,y) and Q(x,y) be (nice) functions. Suppose that we know the following:

(1) The area of R is 10.

(2) The length of C is 3pi.

(3) The function dP/dy – dQ/dx is equal to 7 at every point in R.

Calculate the line integral of P dx + Q dy around the curve C.

I gave full credit plus bonus points to the calculus student who pointed out that my region can’t exist due to the isoperimetric inequality!!

]]>The discussion about asking students for examples took me back to some of the discussion we had at Holt about doing the same thing. I think it is a very powerful tool that teachers don’t use enough. I know I was guilty of that. I do know that when asking students for examples the formation of the question the teachers ask is important so students have a clear idea of what they are expected to do but not so leading that they all just give back the same thing. Oftentimes the question is so open ended that the students just stare back with no idea of what to do. I found the inference that doing so it would save the teacher time a little misleading. I know I often spent more time trying to phrase the question appropriately and then choosing the good examples from the student work to use as the followup. I am not saying it was not worth the time but it is not a time saver. Just a better use of the teacher’s time.

I hope you find my response helpful. Please feel free to share my response with others if you find it helpful.

Mike

]]>I love Bryan Stevenson and EJI, and I loved reading about your experience. Thank you for sharing it.

]]>I would like to teach online if possible because I want to provide some type of service to people locked up. I know it will be a challenge and I know many do not have the privileges and resources I had, but demonstrating a sliver of faith and encouragement might help.

I have taught in underfunded schools and charter schools with at-risk students, and I have worked really well with most students.

]]>Only by moving away from contests was I able to view math as fun, interesting, and full of exploration. I imagine many other students feel the same.

]]>Thank you Sylvia! I’m delighted, if unsurprised, to learn of philosophical work on these types of objects. I’m excited for the opportunity to have consideration of these teaching questions be informed a metaphysics lens in addition to a cognitive psychology lens as in the references discussed in note [10].

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