Excerpts from discussion threads on the online MyNCTM community

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### Robin A. Ward and Jennifer Albritton

This article includes an original artwork using geometry. Art such as this can foster understanding and appreciation of fundamental concepts across fields.

### Keith Nabb, Jaclyn Murawska, Jessie Doty, Annie Fredlund, Stewart Hofer, Casie McAllister, Sami Miller, Zoe Nassif, Savannah Pearson, Abby Pike-Nobile, and Emma Welch

Teachers will inevitably encounter mathematical problem contexts that suggest mainstream views, incorporate deficit language, or make inequities visible. This project reports on a small intervention in which prospective elementary teachers were asked to rewrite a mathematics problem exercising the cultural competence needed in both daily teaching and the critical examination of curricular documents.

### Allison W. McCulloch, Nina G. Bailey, Kristen Fye, and Gayle Scott

In this article, we examine the ways in which the creation of a third space can bridge the divide between coursework and practice for preservice secondary mathematics teachers (PSTs) taking a technology, pedagogy, and content course. A university-based instructor partnered with two high school teachers to create a space in which PSTs draw upon and use both academic and practitioner knowledge while creating technology-based tasks for high school students to use. Our results revealed increased focus on pedagogical decisions in areas such as technology-task design and questioning techniques. The data also indicate that the success of this collaboration was connected to fair distribution of work, feeling valued, and personal benefit and challenges centered on maintaining rejection of hierarchy.

### Cara Djonko-Moore

Early childhood practitioners consider practical strategies to innovate mathematics through cultural responsiveness and funds of knowledge.

### Christopher R. Rakes, Rebecca J. Kirvan, and Ashley Witkowski

Students explore equivalent radical expressions using the relationship between area and side lengths of a square.

### Ryan Gillespie, Julie M. Amador, and Abraham Wallin

Rigorous assessments might be stripping students of their ability to know before a classroom assessment whether their learning is proficient.

### Ceire Monahan, Mika Munakata, and Ashwin Vaidya

Students explore probability in an open‐ended environment by participating in two activities that employ Monte Carlo simulations.

### Mervenur Belin and Gülseren Karagöz Akar

The understandings prospective mathematics teachers develop by focusing on quantities and quantitative relationships within real numbers have the potential for enhancing their future students’ understanding of real numbers. In this article, we propose an instructional sequence that addresses quantitative relationships for the construction of real numbers as rational number sequences. We found that the instructional sequence enhanced prospective teachers’ understanding of real numbers by considering them as quantities and explaining them by using rational number sequences. In particular, results showed that prospective teachers reasoned about fractions and decimal representations of rational numbers using long division, the division algorithm, and diagrams. This further prompted their reasoning with decimal representations of rational and irrational numbers as rational number sequences, which leads to authentic construction of real numbers. Enacting the instructional sequence provides lenses for mathematics teacher educators to notice and eliminate difficulties of their students while developing relationships among multiple representations of real numbers.

### Zandra de Araujo, Samuel Otten, Wenmin Zhao, Jessica Kamuru, and Jaepil Han

Encourage student collaboration in problem solving by altering the *who, when*, and *what* of the homework videos you use in flipped lessons.