February 07, 2014
Physical (also known as concrete) manipulatives like the abacus or blocks have been basic tools of the math teaching trade since time immemorial. The ancient Romans and Greeks conceived of counting boards and eventually the abacus, and the Chinese abacus that came into use centuries later is thought to be an adaptation of the Roman version. Closer to our time, Italian educator Maria Montessori brought manipulatives to early childhood learning over a hundred years ago with great success. Today, the National Council of Teachers of Mathematics (NCTM) recommends the use of manipulatives to teach math at all grade levels to teach from NCTM standards: problem solving, communicating, reasoning, connections, and estimation.
Why do we need manipulatives to teach math?
An ancient Chinese proverb says, “I hear and I forget. I listen and I understand. I do and I remember.” There is wisdom in the proverb, because educational research suggests that the most valuable learning happens when students actively construct their own mathematical understanding. The essential abstract concepts can be difficult for some students, especially younger learners. Manipulatives can make an enormous contribution to sense-making abilities for all students—and have been shown to be of special benefit to students who are high-risk, learning disabled, or with limited English proficiency to understand the symbolic language of math.
With online virtual manipulatives readily available to bring math concepts to life, are they ‘better’ than physical manipulatives?
How are physical and virtual manipulatives defined?
Physical, or concrete, manipulatives are physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics to introduce math concepts. It’s important for children to have a variety of materials to manipulate and the opportunity to sort, classify, weigh, stack, and explore to help them construct their own mathematical thinking and knowledge with Cuisenaire Rods, Base 10 Blocks, and Fraction Circles.
Virtual manipulatives are cognitive technological tools that are a dynamic visual representation of physical manipulatives which can be operated through a computer mouse (or a finger on a tablet) to slide, flip, and turn just like a three-dimensional object. Virtual manipulatives give learners the same opportunity to make meaning and see relationships as the result of their actions, just like a physical manipulative. Although virtual manipulatives have some similarities with their physical manipulative counterparts, as cognitive tools virtual manipulatives have unique characteristics that go beyond the capabilities of physical manipulatives.
What are the differences between physical and virtual manipulatives?
There is some amount of effort on the part of the student to extract the mathematical ideas from interacting with physical manipulatives (like blocks). In contrast, built-in constraint systems in many virtual manipulatives provide support for sense-making, and make mathematics ideas more explicit as the student interacts with the tool. Physical manipulatives don’t provide specific and directed feedback and interaction, while virtual tools react to the learner’s actions, provide prompts, and guidance that help the user focus on the mathematics in the task.
Some virtual manipulatives can be altered, including changing the shape of the onscreen object or marking the object with mathematical notations. Virtual manipulatives are readily available with unlimited access to many copies of an electronic object through the click of the mouse—you never ‘run out’ of blocks—a mouse click can add more blocks. Another plus is availability, because the online environment allows 24/7 access to virtual manipulatives, whether they are free or part of a specific purchased program.
There’s also the ‘fun factor” that engages students: students like using technology, find manipulatives easy to use, and enjoy using the computer to explore ideas and information.
What does research say about the use of virtual vs physical manipulatives?
Studies by Suh and Moyer and Steen, Brooks, and Lyon indicate higher achievement and increased flexibility in learning, and increased motivation and time on task. Both types of manipulatives have no meaning in and of themselves. That’s why Judy Donovan, in a Tech and Learning article, notes that while virtual manipulatives provide some support for individual student use, as with physical manipulatives, students benefit from teacher guidance to help them use the manipulative correctly and connect to the underlying math.
A 2011 study, Virtual vs. Concrete Manipulatives in Mathematics Teacher Education: Is One Type More Effective than the Other? found that incorporating both types of manipulatives into the instruction of prospective mathematics teachers not only helps them to build their own conceptual understanding but also provides them with sound pedagogical strategies for use with their future students. They also found that concrete manipulatives appear to be more effective for building pre-service teachers’ own conceptual understanding, with virtual manipulatives at hand to reinforce those concepts.
Should you use both to improve teaching and learning math?
In the same research study, there was a definite advantage to incorporating both types of manipulatives. The order of manipulative use appears to impact the development of conceptual understanding and the students’ ability to transition to abstract algorithms. Using concrete, followed by virtual manipulatives, is recommended by the researchers. Once conceptual understanding is brought about using concrete manipulatives, the subsequent use of virtual manipulatives seems to facilitate bridging to the abstract. It’s a familiar story: teachers need to make math learning more accessible, engaging, and effective with every tool that’s available, particularly with their own skills.