An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least $$n^2/4 - O(n)$$ ordinary circles. Moreover, we determine the exact minimum number of ordinary circles for all sufficiently large n and describe all point sets that come close to this minimum. We also consider the circle variant of the orchard problem. We prove that P spans at most $$n^3/24 - O(n^2)$$ circles passing through exactly four points of P. Here we determine the exact maximum and the extremal configurations for all sufficiently large n. These results are based on the following structure theorem. If n is sufficiently large depending on K, and P is a set of n points spanning at most $$Kn^2$$ ordinary circles, then all but O(K) points of P lie on an algebraic curve of degree at most four. Our proofs rely on a recent result of Green and Tao on ordinary lines, combined with circular inversion and some classical results regarding algebraic curves.

Personalized support for students is a gold standard in education, but it scales poorly with the number of students. Prior work on learnersourcing presented an approach for learners to engage in human computation tasks while trying to learn a new skill. Our key insight is that students, through their own experience struggling with a particular problem, can become experts on the particular optimizations they implement or bugs they resolve. These students can then generate hints for fellow students based on their new expertise. We present workflows that harvest and organize students' collective knowledge and advice for helping fellow novices through design problems in engineering. Systems embodying each workflow were evaluated in the context of a college-level computer architecture class with an enrollment of more than two hundred students each semester. We show that, given our design choices, students can create helpful hints for their peers that augment or even replace teachers' personalized assistance, when that assistance is not available.