Borel circle squaring

We give a completely constructive solution to Tarski's circle squaring problem. More generally, we prove a Borel version of an equidecomposition theorem due to Laczkovich. If $k \geq 1$ and $A, B \subseteq \mathbb{R}^k$ are bounded Borel sets with the same positive Lebesgue measure whose boundaries have upper Minkowski dimension less than $k$, then $A$ and $B$ are equidecomposable by translations using Borel pieces. This answers a question of Wagon. Our proof uses ideas from the study of flows in graphs, and a recent result of Gao, Jackson, Krohne, and Seward on special types of witnesses to the hyperfiniteness of free Borel actions of $\mathbb{Z}^d$.Comment: Minor typos correcte

Impact of a Science Methods Course on Pre-Service Elementary Teachers\u27 Knowledge and Confidence of Teaching with Scientific Inquiry and Problem-Based Learning

The purpose of this study was to measure the impact of an elementary science methods course on pre-service teachers\u27 knowledge and confidence of teaching with inquiry and problem-based instructional strategies. Changes in pre-service teachers\u27 knowledge and confidence were measured before and after completing the course activities using a pilot survey entitled Science Pedagogical Content Knowledge & Confidence (PCKC) Survey. An integrated lecture/laboratory elementary science methods course engaged participants with hands-on activities designed to increase their pedagogical content knowledge: including theory, planning and implementation of inquiry, and problem-based learning. The results indicated that pre-service teachers\u27 knowledge and confidence improved as a result of enrollment in the elementary science methods course. This article validates reform movements to incorporate scientific inquiry and problem-based learning into coursework

Measurable realizations of abstract systems of congruences

An abstract system of congruences describes a way of partitioning a space into finitely many pieces satisfying certain congruence relations. Examples of abstract systems of congruences include paradoxical decompositions and $n$-divisibility of actions. We consider the general question of when there are realizations of abstract systems of congruences satisfying various measurability constraints. We completely characterize which abstract systems of congruences can be realized by nonmeager Baire measurable pieces of the sphere under the action of rotations on the $2$-sphere. This answers a question of Wagon. We also construct Borel realizations of abstract systems of congruences for the action of $\mathsf{PSL}_2(\mathbb{Z})$ on $\mathsf{P}^1(\mathbb{R})$. The combinatorial underpinnings of our proof are certain types of decomposition of Borel graphs into paths. We also use these decompositions to obtain some results about measurable unfriendly colorings.Comment: minor correction