Stochastic analysis of average-based distributed algorithms

We analyze average-based distributed algorithms relying on simple and pairwise random interactions among a large and unknown number of anonymous agents. This allows the characterization of global properties emerging from these local interactions. Agents start with an initial integer value, and at each interaction keep the average integer part of both values as their new value. The convergence occurs when, with high probability, all the agents possess the same value, which means that they all know a property of the global system. Using a well-chosen stochastic coupling, we improve upon existing results by providing explicit and tight bounds on the convergence time. We apply these general results to both the proportion problem and the system size problem.

Stop beating the donkey! A fresh interpretation of conditional donkey sentences.

We propose a new approach to conditional donkey sentences that allows us to face successfully the often called proportion problem. The main ingredients of the proposal are van Benthem's generalized quantifier approach to conditionals (van Benthem, 1984), and Barwise's situation semantics (Barwise, 1989). We present some experimental data supporting our proposal.

Revisiting Mr. Tall and Mr. Short

While students are solving a proportion problem, their work in a measure space will enable teachers to take the measure of their thinking.

Posing Problems: Two Classroom Examples

IN ONE NCTM YEARBOOK TITLED PROBLEM Solving in School Mathematics, Butts (1980) reminded teachers of the importance of posing problems properly. What happens if we fail to do so and pose a problem that cannot be solved? This embarrassing moment can be made profitable if we ask students to help fix the problem. Here we share two lessons in which students help teachers pose problems. In the first lesson, fifth graders found a mistake in a proportion problem, and the teacher asked students to help her pose it properly. In the second, a geometry problem with insufficient information taken from an eighth-grade mathematics test was posed intentionally to future teachers in a teacher-training college as a problem-posing challenge. We first describe the two lessons and then close with suggestions on having learners help teachers pose problems properly.