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Home Education Voice of the Educator
By Maeve Mihan and Carol Buckley
Much has been written about changes in math instruction over the last decade, such as implementation of Common Core State Standards, an emphasis on how we ask questions, making real connections for students and encouraging invented strategies and independent thinking. As we’ve developed a better understanding of how students learn mathematics, we’ve seen the substantial impact concrete manipulatives have had on the development of numeracy, understanding and critical thinking with math.
Traditionally, students were taught singular solutions to math problems and expected to memorize facts. But this kind of instruction does not give students a conceptual understanding of math, making more advanced concepts challenging.
The National Math Advisory Panel recommends the use of concrete manipulatives to promote a deeper conceptual understanding of abstract representations in mathematics through the widely known concrete-representation-abstract framework (doing, seeing, symbolic) has proven successful. The concrete stage — where manipulatives are used — has many benefits for students across all grades. It allows them to visualize and manipulate/experience the problems they are solving and connect conceptual ideas to procedural understanding.
By incorporating concrete manipulatives into our primary and secondary classes, math can become a more meaningful and worthwhile experience for our students.
Teachers can use manipulatives to teach a variety of mathematical concepts. They are available in many forms, such as base 10 blocks, counters, cubes and pattern blocks, as well as inexpensive alternatives, such as paper clips, coins, buttons or dried beans.
Manipulatives as a teaching tool should not be limited to elementary math lessons, as students at the middle and secondary level also benefit from them when new concepts are introduced or previous concepts are explored at a deeper level. Middle-school students who learned to solve algebra transformation equations using concrete manipulatives outperformed peers who learned via traditional instruction.
Middle- and high-school students can use playing cards to practice adding, subtracting or multiplying integers. They can coordinate plane models to practice plotting points or equations. Connecting paper strips with a brad, making connections at different places and forming quadrilaterals can help these older students see how the angle measure changes depending on the length of the side. Virtual manipulatives easily found online can engage students too.
Concrete manipulatives are great because they give students choice in how they solve math problems. One elementary-age student may prefer to use connecting cubes to do an addition problem, while another student may prefer to use a 10 frame. Providing students choice increases motivation and excitement for learning.
Concrete manipulatives are not used often enough to help children to develop an understanding of basic number concepts. Teachers should always have various manipulatives available to young students. One easy way to do this is by providing each student with a toolbox filled with different manipulatives they can use at any time.
Once students have a deep understanding of the concept, they will be less dependent on the use of manipulatives to process a problem and can move on to use pictures to represent the problem and solution. Teachers should model this for students so they soon can generate their own drawings to represent problem solutions. Later, students will be ready to tackle the problems using the abstract form: numbers and symbols.
The amount of time children spend in each phase is highly individual. Students should be allowed to revisit the use of concrete manipulatives with each new concept being taught. For example, students might have matured to the abstract stage for addition and subtraction of whole numbers but would likely benefit from manipulatives when addition and subtraction of integers are introduced.
Teachers should support students in each step of the CRA process. In the concrete and representation stages, the abstract form — symbols and numbers — can accompany their work. For example, while students might add three red counters and two yellow counters to get five counters, teachers can show the abstract 3 + 2 = 5.
Using manipulatives to understand a math concept does take time. However, the time typically is recovered with the fluency and automaticity gained when children have an in-depth understanding of the concept.
Maeve Mihan is a junior pre-kindergarten through fourth-grade elementary education major and Carol Buckley is an associate professor of mathematics at Messiah University in Pennsylvania.
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