Manipulatives in Math
Manipulatives include objects in another space that help support student learning. The term can encompass blocks, puzzles, geometric shapes, math counters, and other materials that students can interact with to aid their understanding of abstract concepts. So, they can be used with both younger and older learners as they navigate different concepts.
Types of Manipulatives
Manipulatives, as mentioned above, can include physical things to manipulate or move around to demonstrate a concept. Blocks, for example, can be used for area models. There has also been a rise in online manipulatives on platforms like Desmos and Geogebra. These virtual manipulatives allow students to interact with visual representations of mathematical concepts on a computer or mobile device.
The advantage of virtual manipulatives is the flexibility and added range of what can be displayed visually. It’s easier to visualize sine and cosine changes on a unit circle in the digital space than in person. However, many teachers prefer physical manipulatives to avoid their students getting distracted by the digital environment and being on screens all day.
Usage & Introduction
Some teachers allow for students to experiment with the manipulatives prior to any instruction to develop a feel for how they work and build intuitive understanding. They allow for the students to “enjoy the new tool without being told what to do with it right away”. Some use them in a more guided way, introducing them alongside direct instruction on the concept. Many “never expect students to use any tool without whole group instruction” and prefer to start with a lesson on how to use the manipulative.
Interactive and hands-on learning
Interactive, hands-on learning transforms abstract mathematical ideas into tangible experiences. When students physically manipulate objects—whether blocks, puzzles, or digital models—they actively participate in the learning process. Playing with materials can help build a visual for how things work and how systems and concepts interact with one another. Manipulatives also help students visualize how outputs change with input, such as how area models change when the dimensions of a rectangle are adjusted.
Aligning with research-recommended frameworks like the Concrete-Pictorial-Abstract (CPA) and similar approaches, manipulatives facilitate a progression from tangible experiences to abstract reasoning. In the Concrete phase, students physically handle objects to grasp mathematical concepts. They then transition to the Pictorial phase, where they use drawings or diagrams to represent these concepts. Finally, in the Abstract phase, students apply mathematical symbols and notation to solve problems. This is similar to the usage of manipulatives, which can be guided through each of these phases.
