By 1953, scientists knew much about DNA, including its chemical components and how they related to each other. There was even X-ray crystallography of the molecule. However, the structure of DNA in three-dimensional space remained a mystery. James Watson and Francis Crick experimented with building wooden models, creating several until they found the one that fit all of what was already known. This work, which relied heavily on the use of physical models, earned them and their colleague, Maurice Wilkins, a Nobel Prize. Their discovery of the physical structure of the DNA molecule was a breakthrough that lead to massive advances in the understanding of genetics and in medicine..

Research has confirmed that physical models aid in learning difficult concepts in molecular biology. We also know that architects use physical models to develop ideas and test hypotheses. Engineers create prototypes to test ideas. Sculptors may use small-scale sculptures—called maquettes—to envision their final work. Across disciplines, hands-on experimentation and the creation of physical representations help experts generate and test ideas, as well as to clarify and communicate their thinking.

Yet when studying mathematics in school, we often relegate the use of physical manipulatives to either our youngest students or to those who are having difficulty understanding a concept. This may be because of a belief that learning with concrete models works best to show “simple” ideas or for lower-order–thinking tasks, while drawing images or writing with symbols is often viewed as a marker of more sophisticated understanding.

Related to this is the unidirectional, linear manner in which educators sometimes use cognitive psychologist and educational theorist Jerome Bruner’s enactive, iconic, and symbolic modes of representation in classrooms. (Teachers often refer to these three modes of representation as concrete, pictorial, and abstract.) Bruner did posit that the usual course of intellectual development starts with engaging with concrete materials before a child can meaningfully engage with pictorial and then abstract representations. However, he also contended that (1) the usefulness of these modes is not exclusively linear, (2) spiraling back to concepts in developmentally appropriate ways was key to deepening understanding, and (3), much as Watson and Crick did when modeling DNA, when studying complex concepts learners at all levels can benefit from returning to concrete materials. In everyday classroom practice, however, it seems that teachers often abandon physical manipulatives entirely as soon as students can work with something written on a page.

A Flexible, Interactive Model

We propose that we look at Bruner’s modes of representation as a flexible, interactive system, not as a fixed, linear progression. Mathematical understanding can take shape using concrete, pictorial, or abstract representations, where understanding doesn’t necessarily “graduate” from one level to another but rather can grow due to the interaction of all three modes. We picture them as vertices on a triangle, connected by bidirectional arrows.

This multi-directional approach to developing understanding aligns with the principles of Universal Design for Learning (UDL)—specifically, the importance of providing multiple means of representation. UDL starts from the assumption that learner variability is the norm, not the exception, and that learning is supported when ideas are presented in multiple ways. When we adopt a flexible, interconnected model that emphasizes connections among concrete, pictorial, and abstract representations, we better support learning for everyone.

Using Manipulatives in the Classroom

Teachers should be very clear about the goals of a lesson as they think about what representations to use. In the case of physical manipulatives, the What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics: Intervention in the Elementary Grades notes, “Choosing representations must be intentional and selective to be effective.” For example, base-ten blocks support modeling of place-value concepts—including whole numbers and decimals. Fraction tiles, fraction towers, and fraction circles help to visualize fractional relationships (see Strategies for Math Intervention Can Benefit All Students for related insights). The aim is to choose tools that highlight mathematics in ways that promote conceptual understanding for all learners.

In addition, it is important to consider each student’s learning strengths and challenges when deciding how to introduce a math concept and the supports to help deepen understanding. For example, a student with strong visual-spatial strengths may be invited to draw a diagram to show their thinking; a tactile learner may benefit from using physical manipulatives to build and explore the structure of a problem; a student who starts with an equation or symbolic notation may find it helpful to deepen their understanding by modeling their thinking with physical manipulatives or visual models. By recognizing and building on each student’s unique learning profile, we can select entry points and offer supports that make mathematical thinking more accessible and meaningful.

Asking oneself questions like the ones below can make lessons more accessible and can position manipulatives as a tool for reasoning and sense-making.

• How can manipulatives help students to understand relationships, structures, and patterns?
• How might different students use concrete, pictorial, and abstract representations to help them learn?
• How will one mode of representation connect to, support, and reinforce the other two to build deeper understanding?

Conclusion

We strongly believe that physical manipulatives can be important learning supports for all students at any stage of mathematics learning. So can contextual pictures, printed images of physical manipulatives, diagrams such as number lines, and mathematical symbols. Of course, facility with symbolic notation is important and necessary. However, when learning or problem solving at any level, all kinds of representations, including concrete and pictorial representations, can deepen thinking about the ideas behind the symbols. Learning can begin with any mode of representation, and the richest understanding develops when learners move flexibly among them. The more we intentionally plan and allow for that movement, the more we support all learners in making meaning of the mathematics.