Investing in Teacher Knowledge is Investing in Student Learning

The Concrete–Representational–Abstract (CRA) model, developed by Jerome Bruner, along with Richard Lesh’s Translation Model, has influenced nearly every aspect of the work we do at Meaningful Math. Bruner’s CRA model emphasizes that students build the strongest understanding of mathematical concepts when they move through three stages of learning: using concrete materials, transitioning to visual representations, and finally working with abstract symbols. Similarly, Lesh’s Translation Model highlights the importance of helping students make connections across different representations of mathematical thinking; such as manipulatives, drawings, number sentences, and real-world situations. Together, these models guide how we create teaching resources, blog posts, podcast episodes, printable materials, professional articles, and professional development workshops. They remind us that learning is not linear and that true understanding comes from helping students bridge ideas across multiple forms and experiences.

The work of Douglas Clements and Julie Sarama on learning trajectories has deeply shaped the way we think about teaching and learning mathematics at Meaningful Math. Their framework breaks down mathematical ideas into developmental progressions, showing how understanding builds over time, and offering practical guidance for supporting students at each stage. You’ll see this influence in how we develop resources that align with these progressions, as well as in our progression videos which visually map how understanding unfolds over time. Their research continues to guide us as we help teachers make math instruction more responsive, intentional, and grounded in how children learn.

Kathy Richardson’s book How Children Learn Number Concepts: A Guide to the Critical Learning Phases has been foundational in shaping our understanding of how math concepts develop over time. Her research outlines the specific stages children move through as they build number sense and emphasizes that math learning is developmental; not something that can be rushed or skipped. At Meaningful Math, this work directly informed the creation of our progression videos. These 3–6 minute animated videos trace how math understanding grows from the earliest concepts to more complex ideas, showing how everything is connected. They’re designed to help educators see how critical each phase of learning is, and how gaps in early understanding can create stumbling blocks down the road. Richardson’s framework helps us show teachers what it really means to build math from the ground up.

At the heart of Children’s Mathematics: Cognitively Guided Instruction is a simple but transformative idea: students’ mathematical thinking should drive instruction. Authors Thomas P. Carpenter, Elizabeth Fennema, Susan B Empson, Megan Loef Franke & Linda Levi show that children enter our classrooms with informal strategies and intuitive understandings that, when recognized and built upon, can lead to powerful learning. You’ll see its influence in our resources that focus on student strategy use and problem-solving development, as well as in our workshops, podcasts, and blog posts that explore what it means to teach responsively. CGI reminds us that meaningful math instruction begins with understanding how students make sense of problems, and using that as the foundation for our next steps.

Clear, intentional teaching starts with understanding how students learn. That’s the focus of Helping Children Learn Mathematics by Robert Reys, Mary Lindquist, Diana Lambdin, and Nancy Smith. Their work emphasizes the importance of building conceptual understanding, fostering mathematical reasoning, and supporting problem-solving from the earliest grades. The book outlines the content knowledge teachers need, along with the instructional practices they must understand and apply to help young learners think deeply and flexibly about mathematics. Their examples of effective questioning techniques, developmentally appropriate strategies, and student work samples have reinforced key principles we carry into our resource development. You’ll also see this influence in the language we use throughout our blog posts, podcasts, and workshops. Their work continues to inspire our commitment to helping teachers teach math in meaningful, student-centered ways.

When students are encouraged to explore, take risks, and engage with challenging tasks, their relationship with math begins to change. That’s the core of Jo Boaler’s research, and a message that resonates deeply with our work at Meaningful Math. In Mathematical Mindsets, Boaler shows that all students are capable of deep mathematical thinking when instruction promotes curiosity, collaboration, and a belief in growth. Her emphasis on open-ended, low-floor/high-ceiling tasks has shaped how we design resources that invite exploration and support access for a wide range of learners. At the same time, her work on mindset and equitable practices informs our blog posts, podcast episodes, and workshops, where we encourage teachers to see mistakes as learning opportunities and to cultivate positive math identities in every classroom. Boaler’s research continues to guide our efforts to make math a subject where all children feel capable, challenged, and engaged.

Marilyn Burns' book, About Teaching Mathematics, is crucial to our work at Meaningful Math. Her work emphasizes that math instruction should be rooted in reasoning, communication, and meaningful problem-solving. Burns advocates for asking open-ended questions, encouraging student thinking, and creating space for mistakes as vital parts of the learning process. Her belief that students need time to explore ideas, discuss their reasoning, and construct their own understanding has influenced many of our podcast topics, teaching resources, and professional development workshops. Her emphasis on listening to student thinking continues to shape how we talk about effective math instruction and helps guide teachers in seeing math through the eyes of their students.

The book Teaching Student-Centered Mathematics by John A. Van de Walle, Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams has had a strong influence on many of the instructional resources we’ve developed at Meaningful Math. Their research emphasizes that students learn best when they are actively engaged in constructing their own mathematical understanding, rather than being shown procedures to memorize. The authors advocate for an “upside-down” approach to math instruction, where students explore, reason, and make sense of problems before being introduced to mathematical symbols and formal algorithms. Their work highlights the importance of using open-ended tasks, math talk, and representations to support deep understanding. This approach has guided the design of our hands-on activities and inspired many of the topics we explore in our podcast episodes.

At Meaningful Math, so much of our work is grounded in both constructivist and sociocultural theories of learning. The Constructivist theory, rooted in the work of Jean Piaget, emphasizes that children build their understanding of mathematics through active exploration and internal sense-making. This guides us to design teaching resources that prioritize hands-on and exploratory-based learning experiences over rote instruction. Sociocultural theory, based on the work of Lev Vygotsky, reminds us that learning is deeply social. It happens through interaction, discourse, and collaboration within a meaningful context. This influences our emphasis on number talks, math talks, and the importance of classroom discourse. Together these theories shape our belief that children learn best when they are active participants in their learning, rather than passive recipients of information.