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Stipends Successfully Swell Circle
Thomas J. Clark

The efficacy of stipends in drawing new teachers to participate in math teachers' circle and encouraging previous participants to attend meetings regularly was investigated in this study. A kickoff event was planned to start the year with more fanfare than usual. Stipends were advertised for teachers who attended at least three meetings. Matched pairs data analysis and survey results were used to investigate the observed increase in attendance.

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Domino Circles
Lauren L. Rose, A. Gwinn Royal, Amanda Serenevy, and Anna Varvak

Creating a circle with domino pieces has a connection with complete graphs in Graph Theory. We present a hands-on activity for all ages, using dominoes to explore problem solving, pattern recognition, parity, graph theory, and combinatorics. The activities are suitable for elementary school students, the graph theory interpretations are suitable for middle and high school students, and the underlying mathematical structures will be of interest to college students and beyond.

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The Impact of Math Teachers’ Circles on Teacher Dispositions toward Inquiry-based Learning: A Comparison between a Three-day and a One-day Summer Workshop
Angela Antonou, Rita M. Patel, and Amanda Harsy

High-quality professional development for K-12 teachers is a critical need for both teachers and their students. For teachers to provide more engaging and powerful learning opportunities for their students, researchers suggest that we provide similar opportunities for teachers. That is, professional development should model high-impact instructional strategies. Math Teachers' Circles provide one such model for this type of training. In this paper, we discuss the impact on participants of a one-day and participants of a three-day Math Teachers' Circle workshop. In particular, we compare how teacher dispositions regarding the teaching of mathematics and inquiry-based learning changed between the workshops.

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A Math Without Words Puzzle
Jane H. Long and Clint Richardson

A visual puzzle by James Tanton forms the basis for a session that has been successfully implemented with various audiences. Designed to be presented with no directions or description, the puzzle requires participants to discover the goals themselves and to generate their own questions for investigation. Solutions, significant facilitation suggestions, and possibilities for deep mathematical extensions are discussed; extensive illustrations are included.

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Math Escape Rooms: A Novel Approach for Engaging Learners in Math Circles
Janice F. Rech, Paula Jakopovic, Hannah Seidl, Greg Lawson, and Rachel Pugh

Engaging middle and high school students in Math Circles requires time, planning and creativity. Finding novel approaches to maintain the interest of a variety of learners can be challenging. This paper outlines a model for developing and implementing math escape rooms as a unique structure for facilitating collaborative problem solving in a Math Circle. These escape rooms were designed and hosted by undergraduate secondary mathematics education majors. We provide possible structures for hosting escape rooms that could translate to a range of settings, as well as reflections and lessons learned through our experiences that could inform practitioners in other settings.

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Dots Explode in Hawaiʻi
Veny Liu and Laurie James

Teachers are wonderful advocates of mathematics for future generations, and they are continually looking for ways to get students more engaged in mathematics. Through visuals and hands-on activities, the Exploding Dots concept can help teachers and students understand many elementary arithmetic and algebra topics. The implemented tasks promote problem-solving by allowing multiple entry points and varied solution strategies. This paper explored this idea beyond drawing clusters of dots by Locking Legos activity. With a thorough understanding of math content, participants in multiple Math Teachers’ Circle of Hawai‘i (MaTCH) meetings expressed confidence in creating and developing meaningful and relevant differentiated learning opportunities, which include teacher candidates through classroom activities and demonstrations. This paper presents these participants' experiences of a simple concept that grew into a mathematical story.

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Mathematical Zendo: A game of patterns and logic
Philip DeOrsey, Corey Pooler, and Michael Ferrara

Mathematical Zendo is a logic game that actively engages participants in pattern recognition, problem solving, and critical thinking while providing a fun opportunity to explore all manner of mathematical objects. Based upon the popular game of Zendo, created by Looney Labs, Mathematical Zendo centers on a secret rule, chosen by the leader, that must be guessed by teams of players. In each round of the game, teams provide examples of the mathematical object of interest (e.g. functions, numbers, sets) and receive information about whether their guesses do or do not satisfy the secret rule. In this paper, we introduce Mathematical Zendo, provide examples of games and rules that have proven to be engaging over testing with hundreds of students and teachers, and discuss best practices for implementation.

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A Gentle Introduction to Inequalities: A Casebook from the Fullerton Mathematical Circle
Adam Glesser, Matt Rathbun, and Bogdan Suceavă

Run for nearly a decade, the Fullerton Mathematical Circle at California State University, Fullerton prepares middle and high school students for mathematical research by exposing them to difficult problems whose solutions require only age-appropriate techniques and background. This work highlights one of the avenues of study, namely inequalities. We cover Engel's lemma, the Cauchy--Schwartz inequality, and the AM-GM inequality, as well as providing a wealth of problems where these results can be applied. Full solutions or hints, several written by Math Circle students, are given for all of the problems, as well as some commentary on how or when to assist students, and details about the pedagogical value of certain problems.