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## Learning Trajectories for Vertical Coherence

Three learning trajectories and their connections show how to promote vertical coherence in PK–12 mathematics education.

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## Puddle Play!

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## Student Engagement with the “Into Math Graph" Tool

We introduce the Into Math Graph tool, which students use to graph how “into" mathematics they are over time. Using this tool can help teachers foster conversations with students and design experiences that focus on engagement from the student’s perspective.

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## Envelope Curves Unify Sinusoidal Graphing

Transferring fundamental concepts across contexts is difficult, even when deep similarities exist. This article leverages Desmos-enhanced visualizations to unify conceptual understanding of the behavior of sinusoidal function graphs through envelope curve analogies across Cartesian and polar coordinate systems.

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## Leveraging Read Alouds for Mathematical Connections

This article presents examples of how early childhood educators (prek-2nd grade) might use their daily read alouds as a vehicle for increasing mathematical talk and mathematical connections for their students.

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## Now: The Metamorphosis of the Educational World

“It's a YouTube World” (Schaffhauser, 2017), and educators are using digital tools to enhance student learning now more than ever before. The research question scholars need to explore is “what makes an effective instructional video?”.

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## Modeling a Bouncing Ball with Exponential Functions and Infinite Series

We modify a traditional bouncing ball activity for introducing exponential functions by modeling the time between bounces instead of the bounce heights. As a consequence, we can also model the total time of bouncing using an infinite geometric series.

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## The Innovative Impact of NCTM Journals

NCTM has provided rich resources through the publication of practitioner journals for decades and is now leading the way once again with a digital first dynamic publication focused on the learning and teaching of mathematics. This is a rich opportunity for teachers to engage, to learn and to go.