Universal Design for Learning in Mathematics Instruction, K-5

© 2025 CAST, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of the Publisher.

ISBN (paperback) 978-1-943085-30-9

ISBN (ebook) 978-1-943085-31-6

Library of Congress Control Number 2024946210

Cover Design: Lindie Johnson

Interior Design and Cover Production: Happenstance Type-O-Rama

Published by CAST Professional Publishing, an imprint of CAST, Inc., Lynnfield, Massachusetts, USA

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To Torin, Aylin, Brec,

Boden, Jack, and Charlie.

May you always embrace

your inner math person.

Love, your moms

Introduction

Sparking Inclusive Practices in Mathematics

There is a classic scene in the Alfred Hitchcock movie To Catch a Thief where the characters played by Cary Grant and Grace Kelly smooch on a terrace overlooking the French Riviera against a background of fireworks. This scene represents sparks flying between the two characters. The metaphor of sparks flying doesn’t always have to be romantic, though. Sometimes, it denotes the physical generation of sparks, implying a forceful collision charged with energy. This interpretation beautifully captures our first meeting, when Ashley’s expertise in equitable, inclusive mathematics instruction collided with Katie’s in Universal Design for Learning (UDL).

No, we didn’t meet on a terrace overlooking the French Riviera, although that would have been amazing! We met at a conference held at the Killington Grand Resort Hotel in Vermont that focused on building more inclusive systems with the Vermont Superintendents Association. Teams from around the state assembled to create strategic plans to be more inclusive and ensure that students, especially those with the most significant support needs, had access to grade-level classrooms with their peers, as mandated by Act 173—a law enacted by the State of Vermont in 2018 to enhance the effectiveness, availability, and equity of services provided to all students who require additional support in Vermont’s schools.

Act 173 is a significant piece of legislation that fundamentally changed how special education is funded and how general education and intervention services are delivered to students with significant support needs to ensure they have equitable access to inclusive and rigorous learning environments with their peers in all content areas. The act is intended to support students with an individualized education program (IEP), students with a 504 plan under the Rehabilitation Act of 1973, and students who have neither an IEP nor a 504 but whose ability to learn is hampered by a disability or by social, emotional, or behavioral needs. Act 173 is also intended to support all students who are English learners and students who read below grade level (Vermont Agency of Education, 2024).

Inclusion is critical in all grades and subjects, and we firmly believe that all students should have access to instruction in classrooms alongside their peers. That being said, some content areas have become more inclusive than others, and one of the areas that creates the biggest hurdles to access and inclusion is—you guessed it—mathematics.

The lack of access to rigorous mathematics education and the importance of removing barriers to that access have been documented for decades. The Algebra Project, founded by civil rights activist and mathematician Robert Moses, provides a case study that highlights the importance of ensuring that all students, especially those historically resilient and institutionally marginalized in math education, gain access early on to critical mathematical concepts. His idea was revolutionary at the time: to treat math as a civil right. Yet, decades later, many students still do not have access to rigorous mathematics instruction. According to the 2022 results from the National Assessment of Educational Progress (NAEP), also known as the Nation’s Report Card, the average mathematics scores for fourth and eighth graders have significantly declined.

The 2022 report highlights that average math scores for fourth graders fell by five points and for eighth graders by eight points compared to 2019, marking the largest decline in mathematics since the assessment began in 1990. Only 36% of fourth graders who completed the assessment and 26% of all eighth graders who completed the assessment were rated at or above the proficient level in mathematics, indicating a lack of proficiency in fundamental mathematical skills necessary for higher-level problem-solving and reasoning (Nation’s Report Card, 2022).

The most devastating findings underscore the growing disparities in achievement between high-performing and low-performing students, suggesting that the most vulnerable populations are the hardest hit. We cannot ignore the devastating reality that many students, particularly those who are historically resilient and institutionally marginalized, continue to face inequitable access to quality mathematics education. We also want you to remember that these statistics do not define our students’ potential. Rather, they underscore the urgent need for change.

Over lunch from the Make-Your-Own-Taco Bar (a delightful nod to the principles of UDL!) at the Killington Grand Resort Hotel, we discussed our current realities, like the results found in the Nation’s Report Card, and our collective passion for working to dismantle the systems that continue to perpetuate these outcomes. Following that conference, we decided to write an article together. We pitched an article to Edutopia, Making Math Accessible for All Students, and were thrilled when it was published. In the article, we wrote:

All students can build mastery in mathematics when the conditions are right. Access to excellent math instruction for all students creates opportunities for higher learning and ultimately better lives. For students to be successful in learning math, they must demonstrate reasoning and sense-making of math concepts.

Specifically, all students need to demonstrate success in algebra to open pathways into college and careers. If algebra is the gatekeeper to future success, we can open this gate by teaching mathematics through the lens of Universal Design for Learning (UDL). UDL is a framework that provides students with opportunities to work toward firm goals through flexible means.

When used in mathematics classrooms, UDL helps minimize barriers that prevent students from seeing themselves as capable problem-solvers with agency as mathematicians. For students to reach their full potential as math people, all learners must participate in meaningful, challenging learning opportunities. (Marlow & Novak, 2022)

As soon as the article was released, our amazing editors, David Gordon and Billie Fitzpatrick, reached out and said, We loved the article. You two need to write a book together. And so, here it is.

This book is for K–5 mathematics teachers to learn more about how to support all students to learn at high levels in mathematics. While we acknowledge the sobering statistics on math proficiency in the United States, it’s important to recognize that this book is not centered on those figures. We hope to provide a resource for teachers to reflect on their current practice and learning experience in math and ask themselves: What can I do to improve the math learning experience for me and my students? How do I create a math learning environment that sends the message that we are all capable mathematicians? What support and additional professional learning would I benefit from to make this happen?

In this text, we combine our knowledge in both UDL and mathematics education to build on the critical work of the Algebra Project and the importance of rigorous mathematics as a civil right. As members of the UDL community—a steadily growing, tight-knit network—we feel incredibly humbled to publish CAST’s first book about UDL focused solely on mathematics. We are driven by a shared belief that a decline in math achievement is unacceptable, especially given the hard work of educators and the clear evidence of what works in mathematics education. As you explore these pages, we invite you to ignite your inner inclusive practice gladiator and discover the math person within each of your students, thereby reshaping your educational landscape to best meet the needs of all learners. Let the sparks fly—marking the beginning of a journey toward equitable education for every student.

Reflection Question

As you embark on reading this book, what specific goals do you have in mind to enhance your teaching practice, especially in making math accessible to all students? Consider sharing your goals with a colleague or on social media to create a dialogue around overcoming educational barriers and fostering inclusivity through UDL.

1

Math Education

Where It Has Been and Where It Needs to Go

We Love Math . . . Now

We want to tell you a story about a student, AE. Like many of us, AE didn’t always identify as someone who loved math. Caught in the wave of tracking that surged after the No Child Left Behind Act of 2001, she was sorted during middle school into a path that seemed predetermined by a single test—one taken while she sat in a row of desks, a #2 pencil in hand, filling out pages that felt more like verdicts than assessments.

AE loved the humanities—literature, history, languages—but math was a different story. After being placed in the basic math track, AE was excluded from higher-level science courses, which were thought to require a level of math proficiency she presumably did not possess. Throughout high school, AE excelled in every subject but math, padding out her transcript with AP courses in arts and humanities.

College required only a minimal math requirement, referred to dismissively as finite math—a box to be checked by those not pursuing STEM fields. This minimal engagement didn’t improve AE’s confidence or