Educational manipulatives for mathematics

Summary: Some educators use objects to engage students’ attention and encourage them to learn sensorially and experientially.

Educational manipulatives are physical, technological, or virtual objects that are intended to help students learn concepts by taking advantage of tactile and visual explorations.

Mathematical tools and technologies are common in mathematics education. Entire companies and sales catalogues are devoted to such mathematical products, and national and state curriculum standards emphasize their importance in schools. There is a rich history of tools and manipulatives in mathematics classrooms, and these have changed over time along with curricular, industrial, and technological needs and innovations. For instance, in the seventeenth century, the slide rule replaced logarithmic tables in scientific calculation and mathematics classrooms but these in turn became obsolete in the twentieth century because of calculators and computers. Educators, including classroom teachers and university researchers, along with professional designers, continue to create and refine manipulatives and research their effectiveness. Some also work for companies to develop or market these products and materials.

History

Two early developers of collections of learning manipulatives included Friedrich Fröbel (1782-1852) and Maria Montessori (1870-1952). Fröbel was a German educational researcher who is also referred to as the “inventor of kindergarten.” He developed a set of manipulative tools called the Fröbel Gifts, which were intended for kindergarten play and learning in the nineteenth century. The fuller development of the manipulatives occurred after Fröbel’s death. The Fröbel Gifts set contained objects such as balls, cubes, tiles, sticks, and framed figures that were built out of toothpicks and peas. They were designed to help young students explore mathematical concepts in two and three dimensions. Some of the surfaces were hung from string in order to highlight their cross-sections and symmetries. One focus of Frobel’s kindergarten philosophy was free play, which was also carried out in different settings with other objects. For example, the Milton Bradley Company, an American game company established in 1860, sold a curvilinear set of pieces that could form a cylinder. In addition to free play, students ultimately learned to draw what they observed. Ideally, children would revisit concepts they learned using the manipulatives in increasingly sophisticated ways as they progressed through school. For example, in 1869, Edward Wiebe, who was an early proponent of kindergarten education in the United States, suggested that children could explore concepts like the Pythagorean theorem, named for Pythagoras of Samos, long before they understood the square of a number. Frank Lloyd Wright acknowledged the influence of Fröbel’s Gifts on his career as an architect. Aspects of Fröbel’s legacy continue to be found in manipulative design and in schools, although they have been greatly modified and adapted.

In the twentieth century, Italian physician and educator Maria Montessori, who is well known for the Montessori method of education, also focused on the importance of manipulatives in classrooms. She developed an integrated set of sensorial learning materials that included cylinders, cubes, rods, circles, triangles, polygons, boxes, and binomial and trinomial cubes. Montessori designed activities with educational outcomes in mind. Her ideas became popular in the United States and are still used in the twenty-first century. Montessori schoolteachers challenge students to arrange objects in specific ways so that the students will uncover concepts.

Examples

There have been a wide number and variety of other educational manipulatives created in the twentieth and twenty-first centuries, including polyhedral dice with varying numbers of sides for studying probability; multiplication blocks; algebra tiles that represented polynomials and polynomial operations; multicolored and interlocking Unifix cubes intended to teach number and operations concepts; pattern blocks for studying tessellations and fractions; tangrams for exploring geometry; and geoboards, which are pegged boards on which rubber bands could be placed and stretched to investigate concepts like perimeter and area. The abacus or counting frame that had been in use since antiquity found its way into U.S. schools in the nineteenth century. While it has mostly disappeared from twenty-first century classrooms, it remains important in a few educational contexts, like in classrooms for visually impaired children. Virtual manipulatives have replaced physical objects in some cases. There are even applets that mimic some of the physical manipulatives like pattern blocks, which teach similar concepts while providing different sorts of tactile and visual stimulation.

Effectiveness

There are diverse opinions regarding the effectiveness of manipulatives in mathematics education. In 2005, mathematician David Klein warned, “Too much use of them runs the risk that students will focus on the manipulatives more than the math and even come to depend on them.” Yet many state standards recommend and even require the use of a dizzying array of manipulatives in counterproductive ways.” In the final report of the National Mathematics Advisory Panel in 2008, the panel cautioned that

Despite the widespread use of mathematical manipulatives such as geoboards and dynamic software, evidence regarding their usefulness in helping children learn geometry is tenuous at best. Students must eventually transition from concrete (hands-on) or visual representations to internalized abstract representations. The crucial steps in making such transitions are not clearly understood at present and need to be a focus of learning and curriculum research.

Developmental psychologists and educators David Uttal, Kathyrn Scudder, and Judy DeLoache noted that

…the sharp distinction between concrete and abstract forms of mathematical expression may not be justified. We believe instead that manipulatives are also symbols; teachers intend for them to stand for or represent a concept or written symbol.

Other researchers and teachers counter the claims that there is insufficient evidence; they cite a vast amount of educational literature and anecdotes regarding the benefits of hands-on activities, software, and manipulatives. Many students report that they enjoy the tactile manipulation. Students may also feel satisfied when they discover or confirm mathematical relationships, and this may help them connect to mathematics. Mathematics educators continue to study the effects of various manipulatives and the potential differences between physical and virtual manipulatives on student learning.

Bibliography

Burns, Marilyn. “How to Make the Most of Math Manipulatives: A Fresh Look at Getting Students’ Heads and Hands! Around Math Concepts.” Instructor 105, no. 7 (1990). http://teacher.scholastic.com/lessonrepro/lessonplans/instructor/burns.htm.

Kidwell, Peggy, Amy Ackerberg-Hastings, and David Roberts. Tools of American Mathematics Teaching, 1800-2000. Baltimore, MD: John Hopkins University Press, 2008.

Klein, David. The State of State MATH Standards. Washington, DC: Thomas B. Fordham Foundation, 2005. http://www.math.jhu.edu/~wsw/ED/mathstandards05FINAL.pdf.

Moyer-Packenham, Patricia. Teaching Mathematics With Virtual Manipulatives, Grades K-8. Rowley, MA: Didax, 2010.

National Mathematics Advisory Panel. “Foundations for Success: Final Report of the National Mathematics Advisory Panel.” March 13, 2008. http://www2.ed.gov/about/bdscomm/list/mathpanel/index.html.

Uttal, David, Kathyrn Scudder, and Judy DeLoache. “Manipulatives as Symbols: A New Perspective on the Use of Concrete Objects to Teach Mathematics.” Journal of Applied Developmental Psychology 18, no. 1 (1997).