How to Support Math Students with the Universal Design for Learning Principles

How Do UDL Principles Apply to the Design of the Cockpit of a Fighter Jet and a Math Lesson

First, let’s provide a bit of context. In this popular Ted Talk video, best-selling author and Harvard professor Todd Rose tells the story of a problem the United States Air Force faced in the 1950s. Despite having good pilots and better planes, they were getting poor results. After ruling out other possible causes of declining performance, such as the pilots themselves, the technology, and the flight instructors, the air force realized that the problem was with the cockpit in its fighter jets. It didn’t fit the pilots. The thinking behind the existing configuration was that a cockpit designed to fit the average-size pilot would accommodate most pilots. As part of the troubleshooting effort, a researcher measured 10 different body dimensions (height, shoulders, torso, etc.) of over 4,000 pilots to determine how many pilots had average measurements of all 10 dimensions. The air force discovered that no one had average measurements of all dimensions; there was no average pilot. In fact, every pilot had what they described as a “jagged size profile”—some measurements were average, some above, some below. As a result, the air force banned designing for the average, instead requiring companies that built the planes to “design to the edges” of pilots’ dimensions, ensuring that the cockpit would fit the variability of pilots’ sizes. The result was a cockpit that accommodates all pilots by providing flexible components such as adjustable seats. 

Just as the air force discovered that there is no average-size pilot, neuroscience research tells us that there is no average learner. As an article by CAST, a nonprofit education research and development organization, explains, no two brains are alike (2018a). Therefore, instruction should not be designed for the “average learner.” Instead, instruction should provide flexible options—just like the adjustable seat in the cockpit of a fighter jet. CAST’s Universal Design for Learning Guidelines (2018b), help educators design instruction that proactively addresses barriers to learning by designing for the predictable variability of all learners. 

Universal Design for Learning in Bluebonnet Learning K–5 Math

The authors of Bluebonnet Learning K–5 Math applied the UDL Guidelines to develop lessons that provide multiple means of engagement, representation, and action and expression. Options that address learner variability are built into the lesson design and suggested at point of use in margin notes. 

Why is the Universal Design for Learning Important?

UDL is a framework based on current research from cognitive neuroscience that recognizes learner variance as the norm rather than the exception. The guiding principles of the UDL framework are based on the three primary networks of the brain.

According to CAST, the UDL framework “offers an overarching approach to designing meaningful learning opportunities that address learner variability and suggests purposeful, proactive attention to the design of goals, assessments, methods, and materials.” (n.d.) The UDL Guidelines provide a set of concrete suggestions organized by the three guiding principles: Engagement, Representation, and Action & Expression. Educators can use the UDL Guidelines to evaluate their goals, assessments, methods, and materials and identify potential barriers to student learning. Teachers can then proactively design instruction that maintains desirable challenges and eliminates unnecessary barriers. Here is an overview of each guiding principle and its instructional implications. 

1. Provide Multiple Means of Engagement: Learners differ markedly in the ways in which they can be engaged or motivated to learn. Use a variety of methods to engage students. For example, provide choice, address student interest, and arrange for students to monitor their own learning, such as with goal setting, self-assessment, and reflection.
2. Provide Multiple Means of Representation: Learners differ in the ways they perceive and comprehend information that is presented to them. Use a variety of strategies, instructional tools, and methods when presenting information and content. 

3. Provide Multiple Means of Action & Expression: Learners differ in the ways they can navigate a learning environment and express what they know. Offer students a variety of strategies and instructional tools and allow for multiple ways to demonstrate new understanding.

How Do You Provide Multiple Means of Engagement in Math?

The following examples show how Bluebonnet Learning K–5 Math lessons apply the UDL principle of Engagement in lessons. 

Provide Options for Recruiting Interest 

According to CAST, “individuals are engaged by information and activities that are relevant and valuable to their interests and goals.” (UDL Guidelines, Engagement, Checkpoint 7.2) In this Grade K lesson, students use name trains—linking cubes representing the letters in their names—to create a class graph and analyze data about who has the letter E in their name. The activity engages students by centering their own names, sparking curiosity and investment through personally meaningful content. A margin note encourages teachers to consider student names and interests when creating categories or survey questions. Tailoring questions in this way provides options for recruiting interest by personalizing and contextualizing the content to learners’ lives. 

Math Research Connection 

“Research supports the effectiveness of integrating students’ interests and experiences into instruction to anchor it in meaningful contexts (e.g., Gersten, 1998; Meyers et al., 2015; Wehmeyer, Palmer, & Agran, 1998). Doing so creates meaningful learning contexts for students, and meaning affects learning in several positive ways, including enhancing memory retrieval,  increasing attention, and helping students with diverse backgrounds and experiences find relevance.” (Allsopp, Lovin, and van Ingen 2018, 211) 

Provide Options for Sustaining Effort and Persistence 

According to CAST, “feedback that orients students toward mastery and that emphasizes the role of effort and practice rather than ‘intelligence’ or inherent ‘ability’ is an important factor in guiding students toward successful long-term habits of mind.” (UDL Guidelines, Engagement, Research for Checkpoint 8.4) In this Grade 2 lesson, students use math drawings to model and solve addition problems involving renaming. When presented with an incorrect solution, students act as “math doctors” to diagnose and correct the error using place value language. A margin note encourages teachers to select students who need additional support to lead this work, highlighting their thinking and effort. Providing opportunities for all students to analyze errors, explain their reasoning, and correct mistakes fosters a growth mindset and provides options for sustaining effort and persistence through relevant, consequential feedback. 

Math Research Connection 

“Effective feedback does not just correct students’ mistakes. Feedback can also help students progress in their understanding, so feedback can be in the form of a question that helps a student focus on a specific idea or connection between ideas. As students explain their reasoning, you should ask questions based on the students’ reasoning rather than the ways you are thinking about the task.” (Allsopp, Lovin, and van Ingen 2018, 235) 

Provide Options for Self-Regulation 

According to CAST, “the ability to self-regulate … is a critical aspect of human development. While many individuals develop self-regulatory skills on their own, many others have significant difficulties in developing these skills.” (UDL Guidelines, Engagement, Guideline 9) In this Grade 5 lesson, students solve decimal multiplication problems using area models and place value strategies. They work with partners to solve problems, compare strategies, and explain their reasoning. A margin note encourages teachers to support students in recognizing their own progress by tracking the skills they master across the module. Encouraging students to reflect on their growth and connect their effort to success provides options for self-regulation by strengthening their belief in their own abilities. 

How do you Provide Multiple Means of Representation in Math?

The following examples show how Bluebonnet Learning K–5 Math lessons apply the UDL principle of Representation in lessons.  

Provide Options for Perception 

According to CAST, “information conveyed solely through sound is not equally accessible to all learners and is especially inaccessible for learners with hearing disabilities, for learners who need more time to process information, or for learners who have memory difficulties. To ensure that all learners have access to learning, options should be available for any information, including emphasis, presented aurally.” (UDL Guidelines, Representation, Checkpoint 2) In this Grade 4 lesson, students decompose fractions using strip diagrams and represent those decompositions with number bonds and equations. A margin note encourages teachers to display visual representations—such as shaded paper strips, number bonds, and annotated diagrams—to support students’ understanding of part–whole relationships. Providing these visual supports alongside verbal explanations offers options for perception by making abstract concepts concrete and reinforcing information in multiple modes. 

Provide Options for Language and Symbols 

According to CAST, “classroom materials are often dominated by information in text. But text is a weak format for presenting many concepts and for explicating most processes. Furthermore, text is a particularly weak form of presentation for learners who have text- or language-related disabilities.” (UDL Guidelines, Representation, Checkpoint 2.5) In this Grade 1 lesson, students solve compare problems with a difference unknown by modeling with linking cubes and drawing strip diagrams. As students read a story problem about how many more letters one character wrote than another, they use concrete manipulatives and visual diagrams to make sense of the quantities and relationships. A margin note suggests connecting students’ cube models directly to the strip diagram with number labels and physical gestures. Presenting the story problem alongside visual and tactile models provides options for language and symbols by making the abstract relationships in the text more comprehensible and accessible, especially for emerging readers or students with language-related learning needs. 

Math Research Connection 

“Schema-based instruction, in which students are explicitly and systematically taught to use visual diagrams and graphic organizers to solve word problems, has been demonstrated to help students develop and utilize cognitive schemas to problem-solve more effectively (e.g., Jitendra, George, Sood, & Price, 2010; Jitendra & Hoff, 1996).” (Allsopp, Lovin, and van Ingen 2018, 192) 

Provide Options for Comprehension 

According to CAST, “learning can be cognitively inaccessible when it requires the ability to select and prioritize among many elements or sources, and where there are no options for individuals who differ in that capability. One of the most effective ways to make information more accessible is to provide explicit cues or prompts that assist individuals in attending to those features that matter most while avoiding those that matter least.” (UDL Guidelines, Representation, Research for Checkpoint 3.2) In this Grade 3 lesson, students analyze two bar graphs—one showing the price of a video game and the other showing how many units were sold over time—to explore the relationship between supply, demand, and cost. Teachers prompt students to compare the graphs and use the visual data to make inferences about price changes. A margin note encourages the use of graphic organizers and modeled examples to help students interpret and connect abstract economic terms to visual data. Visual supports like labeled graphs and guided questions provide options for comprehension by drawing attention to key patterns and reinforcing conceptual understanding through multiple representations. 

Math Research Connection 

“Graphic organizers can be effectively used to help students explicitly make connections between mathematical terms and the mathematical ideas to which they relate. Graphic organizers are especially helpful for struggling learners because they provide a visual schema that helps students make associations between words and ideas in their mind’s eye. This reduces the cognitive demand on students; it is easier than making these associations abstractly without such supports.” (Allsopp, Lovin, and van Ingen 2018, 176–77) “When teachers utilize visuals to provide students with a cognitive framework within which to attach both prior and new mathematical knowledge, they provide students a scaffolded way to build important mathematical connections.” (210) 

How Do You Provide Multiple Means of Action & Expression in Math?

The following examples show how Bluebonnet Learning K–5 Math lessons apply the UDL principle of Representation in lessons. 

Provide Options for Physical Action 

According to CAST, “learners differ widely in their capacity to navigate their physical environment. To reduce barriers to learning that would be introduced by the motor demands of a task, provide alternative means for response, selection, and composition.” (UDL Guidelines, Action & Expression, Checkpoint 4.1) In this Grade 2 lesson, students use counters and number lines to model and solve division problems through repeated subtraction. As students draw and label jumps on open number lines, a margin note suggests offering pre-partitioned number lines or inviting students to describe what to draw to a partner instead of drawing themselves. These supports help reduce fine motor demands, allowing students to engage with the math concept without being hindered by the act of drawing. Providing alternatives for how students represent their thinking offers options for physical action.

Provide Options for Expression & Communication 

According to CAST, “It is important to welcome and encourage a variety of media for expression. Such variety reduces media-specific barriers to communication among learners with disabilities, honors forms of communication that have historically been devalued, and increases the opportunities for every learner to develop a wider range of expression in a media-rich world.” (UDL Guidelines, Action & Expression, Checkpoint 5.1) In this Grade 4 lesson, students solve problems involving mixed units of length and are encouraged to represent their thinking in a variety of ways, including written explanations, partner discussions, and visual models like number bonds and diagrams. A margin note suggests providing scaffolds such as visual aids or allowing students to choose between different solution methods. Offering opportunities to share ideas using multiple media supports diverse learners and provides options for expression and communication that extend beyond traditional written formats. 

Provide Options for Executive Functions 

According to CAST, “learning can be inaccessible when it requires planning and strategy development, and where there are no options for individuals who differ in such executive functions. Young children, older students in a new domain, or any student with one of the disabilities that compromise executive functions (e.g., ADHD, ADD, Autism Spectrum Disorders) often are weak at planning and strategy development and impulsive trial and error dominates their learning.” (UDL Guidelines, Action & Expression, Research for Checkpoint 6.2) In this Grade 1 lesson, students explore strategies for solving addition problems by decomposing addends to make ten. After solving problems, students discuss and compare strategies with a partner before sharing with the class. A margin note prompts teachers to provide this partner talk opportunity to support students who are still developing their problem-solving process. Hearing multiple solution paths and explaining their own thinking gives students a structured way to reflect, revise, and plan—offering options for executive functions by supporting strategy development and metacognitive thinking. 

Math Research Connection 

“Many students require support and guidance from their teachers to develop metacognitive awareness about mathematics. Such support may come in the form of modeling strategies for approaching and solving problems; making one’s own thinking processes transparent through thinking aloud; utilizing visuals (e.g., graphic organizers) to demonstrate how mathematics ideas connect; and providing opportunities for students to tryout problem-solving strategies in risk-free contexts, and to discuss their level of effectiveness through discourse and feedback, for the purpose of improvement.” (Allsopp, Lovin, and van Ingen 2018, 208) 

What is the Importance of Universal Design for Learning in Math?

So back to our original question, how is a Bluebonnet Learning K–5 Math lesson like the cockpit of a fighter jet? Just as the United States Air Force provided options to address the variability of their pilots, Bluebonnet Learning K–5 Math  lessons provide options to address the variability of learners. Rather than designing for the illusory average student, Great Minds® prioritizes learner variability, recognizing that average is a myth and variability is the norm.  

Although the concept of UDL has roots in special education, UDL is for all students. When instruction is designed to meet the needs of the widest range of learners, all students benefit. This is especially important since, according to the National Center for Learning Disabilities report The State of Learning Disabilities: Understanding the 1 in 5, “1 in 5 children in the U.S. have learning and attention issues, but only a small subset are formally identified with a disability in school … while millions of children with learning and attention issues are not formally identified.” The report recommends policy changes, one of which is to “drive innovation for effective teaching and learning.” A key aspect identified to accomplish this change is to use UDL to reach every student. As the report states, “Children with learning and attention issues are as smart as their peers and, with the right support, can achieve at high levels” (Horowitz, Rawe, and Whittaker 2017). 

Learning Acceleration for All: Planning for the Next 3–5 Years, a recent report by TNTP, an education nonprofit, cautions school systems to recognize and address pitfalls that may hinder their long-term learning acceleration strategy. One of these pitfalls is “centering on the mythical ‘average’ student rather than considering how to plan for high expectations for a diverse set of learners” (2021, 19).” TNTP recommends that school systems work “in partnership with parents, teachers, students, and community partners to develop a comprehensive education support plan that will provide equitably high-quality academic experiences to all of [their] students” (6).” A critical component of a high-quality academic experience for students is a high-quality curriculum that is explicitly designed to attend to the needs of all learners in its instructional design, not just the “average” learner because the average does not exist. Each student arrives to school with their own strengths, areas for growth, and approach to engaging with academic content. School systems can be confident when selecting Bluebonnet Learning K–5 Math that the curriculum materials were designed to address learner variability from the outset.