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Math Manipulative Flip Chart

 

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Today’s classrooms are comprised of students with diverse needs. Although classrooms are filled with students representing various learning styles, abilities, and needs, student expectations are the same. No one can be certain what mathematics students will need in order to succeed in the future. However, educators must recognize that today’s students must gain knowledge and develop attitudes that will enable them to become lifelong learners. It appears that conceptual understanding of mathematics is essential and methods that enhance this understanding should be included in a mathematics curriculum. The literature reviewed for this document indicates that manipulative materials should be an integral part of mathematics instruction beginning in the elementary grades.

Math manipulatives are concrete objects that allow students to explore and demonstrate mathematical concepts. With manipulatives, children are actively engaged in doing mathematics and in making meaning. Students can see, touch, manipulate, and move manipulatives in order to model a concept in mathematics. This multisensory approach requires active involvement and addresses the learning styles of multiple students. When implemented in a thoughtful manner, students can use math manipulatives as strategic tools to conceptualize and solve problems, moving from a concrete level to an abstract level of thinking. The effective use of math manipulatives lays an experiential foundation that promotes retention, application, and extension of math concepts (Bellonio, 2012). 

The Math Manipulatives Flip Chart provides engaging ideas and useful information to help teachers in grades 3-5 plan valuable learning activities using manipulatives. This viable resource helps teachers guide students in using manipulatives to build, strengthen, and connect various representations of mathematical ideas. This educational resource for teachers outlines how manipulatives can be used to teach concepts across the strands of the mathematics curriculum, so that manipulatives are not limited to teacher demonstrations. Grouws and Cebulla (2000) note that students must use manipulatives meaningfully and see the relationship between these materials and the corresponding mathematics concept. Furthermore, Grouws and Cebulla emphasize the importance of using manipulatives to focus on thinking about the relationships of the representation to a mathematic concept rather than focusing on memorization of a concept. The Math Manipulatives Flip Chart provides support to teachers in making decisions related to why and how to use manipulatives effectively in the classroom, expanding the knowledge base of teachers for teaching mathematical concepts by using more than one or two types of manipulatives. 

The use of manipulatives in mathematics instruction is a common hands-on practice as is the use of textbooks. Findings show that long-term usage of manipulatives has a positive effect on student achievement by allowing students to use concrete objects to observe, model, and internalize abstract concepts (Sowell, 1989; Ruzic and O’Connell, 2001a). Manipulatives enable students to construct cognitive models for abstract mathematical ideas and processes. In addition, manipulatives provide students a common language with which to discuss their mathematical models with teachers and other students.

Cognitive processing of concepts is strongly impacted when manipulatives are a regular component in the mathematics curriculum. Hartshorn and Boren (1990) found that manipulatives help students transition from intuitive to logical thinking; then move from concrete to abstract. Findings by Sutton and Krueger (2002) note that students who have opportunities to use manipulatives report that they are more interested in math. Furthermore, these researchers state that long-term interest in mathematics translates to increased mathematical ability. Thus, student interest and enjoyment in mathematics appear to increase when students actively and purposefully interact with manipulatives as mathematical concepts are introduced and reinforced.

From earliest history, people have used objects and counters to solve mathematics problems. Swiss social reformer and educator Johann Pestalozzi was an advocate of manipulatives in the 19th century. Saettler (1990), an educational historian, discovered evidence that supported the use of manipulatives as concrete aids to understand number sense. Toward the beginning of the 20th century, Ward (1971) found that Montessori also supported the use of concrete, self-directed learning experiences with developmentally appropriate manipulatives.  

In the 1960s and 1970s, the use of manipulatives in elementary classrooms grew more prevalent and studies were designed to determine the effectiveness of manipulatives. Research continues today at all grade levels and with different abilities of students, examining the impact of various manipulatives (concrete and virtual) on student outcomes as well as how teachers use and misuse manipulatives, including the effects on the environment in math classrooms. Much research has shown that manipulatives more times than not have a positive effect on student achievement and mathematical learning when compared to more traditional instructional methods which emphasize the use of worksheets and computational fluency (Bisio, 1971; Fennema, 1972; Suydam and Higgins, 1977; Driscoll, 1981; Parham, 1983; Sowell, 1989; Cramer et al., 2002). Some researchers have questioned whether manipulatives have value during mathematics instruction (Friedman, 1978; Raphael and Wahlstrom, 1989). Based on eighteen dissertations, Friedman discovered that the results of ten studies showed no significant differences in achievement between manipulative based instruction as compared to traditional instruction without manipulatives. In four other dissertations, mixed results were reported. Therefore, Friedman concluded that manipulative-based instruction may show some advantages over non-manipulative based instruction, but the overall findings appear inconclusive. 

An analysis to determine the effectiveness of three different methods for teaching addition and subtraction of fractions with like denominators was investigated by Bisio (1971). In the first method, no manipulatives were used by teachers or students;

in the second method, manipulatives were only used by teachers for demonstration of concepts; in the third method, both teachers and students utilized manipulatives. Bisio concluded that the use of manipulatives by teachers in method 2 was as effective as the use by students in method 3. However, both approaches were more effective than method 1 in which no manipulatives were employed.

Fennema (1972) investigated the use of Cuisenaire rods to teach as compared to more traditional approaches. The results of her research generally supported the use of Cuisenaire manipulatives for first-graders, but the value of using the Cuisenaire rods for second- and third-graders was less conclusive. Fennema recommended that teachers use manipulatives to teach mathematics and gradually decrease usage when students can successfully transition from manipulative-based conceptual representations to abstract representations (symbols and words).

Suydam and Higgins (1977) conducted a meta-analysis of 40 research studies about the use and effectiveness of manipulatives on student achievement in math. The results produced these various findings. In 60% of the studies, manipulatives seemed to have a positive effect on student learning; 30% of the studies demonstrated no effect on achievement; and the remaining 10% revealed significant differences favoring the use of more traditional instructional methods with no manipulatives. Using 64 research studies, Parham (1983) also analyzed the use of math manipulatives at the elementary level. Her findings show significant positive differences in the achievement of students who had used manipulatives as part of their math instruction as compared to students who had not. In summary, the meta-analysis indicates that students who use manipulatives to learn mathematical concepts, no matter the area of mathematics, outperform students who receive no exposure to manipulatives (Parham, 1983; Sowell, 1989). 

In another study regarding the use of Cuisenaire rods with third-grade learning disabled students, Marsh and Cooke (1996) analyzed the effects of using manipulatives to identify the correct operations to use when solving math word problems. After the utilization of manipulatives, the results yielded statistically significant improvements in students’ abilities to identify and use the correct operations when solving the presented word problems. 

In a study of 1,600 fourth- and fifth-graders, Cramer and his colleagues (2002) compared the achievement of students using a commercial curriculum for teaching fractions to the achievement of students in a curriculum with an emphasis on the usage of manipulatives. Students participating in the manipulative-based curriculum demonstrated statistically higher mean scores on posttests and retention tests. Cramer noted the importance of students interacting with manipulatives over an extended period in order to form the mental images that would lead to understanding of mathematical concepts.

Suydam and Higgins (1977) found that one year or longer of mathematics instruction with concrete models increased achievement. Suydam indicated that short-term investigations yielded no difference between manipulative and non-manipulative student groups. Earlier, Thornton and Wilmot (1986) reported that manipulatives are effective tools for helping students with learning disabilities understand mathematics. Spear-Swerling (2006) cautions against instruction without appropriate teacher guidance and instruction. She reports the necessity of explicit and systematic instruction of struggling students including those who are learning disabled. Many students benefit through the use of visual representations, pictorial representations, and concrete or virtual manipulatives. However, all students must be closely monitored to determine the proper usage and appropriate integration of manipulative-based instruction.

The effects of gender were also a variable investigated with manipulative-based approaches. While Klahr et al. (2007) found no significant impact in results, the study reported boys outperformed girls on one measure and girls scored higher on another measure. Thus, Klahr reported no significant gender difference in learning gains of students exposed to manipulative-based instruction. 

Based on a review of 14 studies, the National Center for Accessing the General Curriculum (Ruzic et al., 2001b) reported that “use of manipulatives compared with traditional instruction typically had a positive effect on student achievement.” This reporting applied to all subpopulations, but was especially relevant for high-risk, learning disabled, and English Language Learners (ELLs). Manipulative materials enable ELLs and students with special needs to observe concepts being modeled as words, as the mere words of their teachers are often not understood by these student groups. Manipulatives provide opportunities for all learners, specifically struggling learners, to build representations and demonstrate knowledge of ideas that they are unable to communicate using symbols or words. The Math Manipulatives Flip Chart offers suggestions on how teachers can use various manipulatives to increase student comprehension, improve student retention, and/or assess mathematical concepts.

Although research findings may be mixed, there is no question that today’s educators are in general agreement that an effective elementary mathematics curriculum should include frequent use of manipulatives. A nationwide survey of 1,000 members of the National Education Association (NEA, 2002 Instructional Materials Survey) indicated the importance of manipulatives in teaching and learning math. The results showed 49% of the elementary grade teacher respondents stated they use manipulatives on a daily basis as part of ongoing instruction. Teachers across the nation were asked to rate the effectiveness of manipulatives and 67% responded that manipulatives were highly effective instructional tools. An increase in response was noted by elementary teachers as 85% of teachers in elementary grades rated manipulatives as highly effective resources.

Learning theory also supports the use of manipulatives as aids in developing mathematical understanding in young children. Piaget was an advocate and often proclaimed that children only begin to understand symbols and abstract concepts after experiencing ideas on a concrete level. Heddens (1986) indicated that students who manipulate (see, touch, and sort) physical objects develop mental images that will help them explain, draw, or represent abstract ideas more clearly than students with limited experiences with manipulatives. In 1960, Dienes concluded that children whose mathematical learning is established with manipulative experiences are more likely to bridge learning gaps in their knowledge and mathematics. The rationale for the development of the Math Manipulatives Flip Chart was firmly established because evidence demonstrates that manipulatives are effective tools in mathematics education for providing children the support needed to move from a concrete to an abstract level of understanding. 

Support is referenced in national and state curriculum standards in relation to math manipulatives. Authorities in the field of mathematics agree that young children learn to understand math concepts best with the use of manipulatives. According to the 1989 publication Curriculum and Evaluation Standards for School Mathematics, the National Council of Teachers of Mathematics (NCTM) strongly emphasizes the importance of manipulatives in math education, specifically at the elementary level so that children are actually ‘doing math’. This recommendation is based on the support found in learning theory as well as classroom educational research. An example of the K–4 standard for number sense and numeration is, “Children come to understand number meanings gradually. To encourage these understandings, teachers can offer classroom experiences in which students first manipulate physical objects and then use their own language to explain their thinking. This active involvement in, and expression of, physical manipulations encourages children to reflect on their actions and to construct their own number meanings. In all situations, work with number symbols should be meaningfully linked to concrete materials.” 

Published in 2000, the revision of the NCTM Standards Principles and Standards for School Mathematics also prescribes increased usage of manipulative materials in mathematics instruction, particularly in the early grades. Seefeldt and Wasik (2006) noted the importance of establishing mathematical development in the early years as indicated by the aforementioned document. Boggan et al. (2010) reported that learning is beneficial when students assume active roles in constructing mathematical meaning. They indicated that meaning is often reached through the utilization of manipulatives. Most states require the use of manipulatives through their specified curriculum. Pictorial representations of manipulatives often appear on state assessments. Thus, as students understand a concept using manipulatives, they need to be taught how to express the learning on paper. Transitioning from manipulatives to paper and pencil tasks is an important and necessary step in mathematics education.

The Product Development Team for Mathematics recognizes the importance of manipulative materials in mathematics education. Providing students with direct, concrete experiences is supported by evidence from the classroom and an understanding of how learning takes place. While children can retain information taught through books and lectures,studies show that deep understanding and the ability to transfer and apply knowledge to new situations require learning that embeds direct, concrete experiences. Manipulative-based instruction establishes a foundation from whence students become independent learners and thinkers.

Raphael and Wahlstrom (1989) and Sowel (1989) found that a relationship existed between student achievement levels and teachers’ experiences and expertise with manipulatives. Mentoring Minds Mathematics Team believes that children cannot learn math by merely manipulating physical objects, thus, the reason for the development of the educator resource Math Manipulatives Flip Chart. As former educators, we know that many teachers believe manipulatives are effective tools, yet not all teachers use manipulatives.  We also recognize that there are some teachers who are unfamiliar with how to correctly use manipulatives.

The Math Manipulatives Flip Chart provides an in-depth look at twelve commonly used math manipulatives in grades 3-5. Tabbed sections of the flip chart provide a brief overview of each manipulative followed by a variety of strategies and activities for using the identified manipulative. The strategies and activities within each tabbed section are arranged topically by the math domain or strand they address: Numeration, Operations and Algebraic Thinking, Fractions, Measurement, Probability and Statistics, and Geometry. Each strategy or activity begins with a title that expresses the main idea followed by a detailed description of how this strategy or activity should be used in classroom instruction. Suggested probing or follow-up questions and extensions are provided. Instructional tips are also offered to encourage teachers to closely monitor student work in order to maintain focus on or discovery of mathematical concepts. Furthermore, tips are given to assist teachers as they help students build bridges from concrete work to corresponding work with symbols. While there appears to be no single best way to teach math although many warrant careful consideration, research does show that using manipulatives in conjunction with other methods can deepen students’ understanding of abstract concepts and optimize quality instruction in mathematics (Grouws and Cebulla, 2000). In conclusion, this literature review demonstrates that manipulatives should be one component of a comprehensive mathematics curriculum. Overall, research indicates that manipulatives positively impact students’ conceptual understanding in mathematics. It is the intent of the Mentoring Minds Product Development Team for Mathematics that the Math Manipulatives Flip Chart is a guide to help teachers appropriately use manipulatives in classrooms to improve math education.

 

Bibliography for Math Manipulatives Flip Chart

Bellonio, J. (2012). Multi-sensory manipulatives in mathematics: Linking the abstract to the concrete. Retrieved April 2012 from Yale-New Haven Teachers Institute. http://www.cis.yale.edu/ynhti/curriculum/units/2001/6/01.06.12.x.html

Bisio, R. (1971). Effect of manipulative materials on understanding operations with fractions in grade V. Dissertation Abstracts International, 32, 833A.

Boggan, M., Harper, S., & Whitmire, A. (2010). Using manipulatives to teach elementary mathematics. Journal of Instructional Pedagogies, 3 (1), 1-10. Retrieved Spring 2012 from

http://www.aabri.com/manuscripts/10451.pdf

Cohen, H. (1992). Two teaching strategies: Their effectiveness with students of various cognitive abilities. School Science & Mathematics, 92: 126–132.

Cramer, K., Post, T., & delMas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33: 111-144. 

Dienes, Z. P. (1960). Building up mathematics. London: Hutchinson Educational Ltd.

Driscoll, M. (1981). Research within reach: Elementary school mathematics. Reston, VA.: National Council of Teachers of Mathematics.

Friedman, M. (1978). The manipulative materials strategy: The latest pied piper? Journal for Research in Mathematics Education, 9: 78-40.

Grouws, D., & Cebulla, K. (2000). Improving student achievement in mathematics. Brussels, Belgium: International Academy of Education and Geneva, Switzerland: International Bureau of Education. Retrieved Fall 2011 from http://www.ibe.unesco.org/fileadmin/user_upload/archive/publications/EducationalPracticesSeriesPdf/prac04e.pdf

Hartshorn, R., & Boren, S. (1990). Experiential learning of mathematics: Using manipulatives. Martin, TN and Charleston, WV: ERIC Clearinghouse on Rural Education and Small Schools.

Heddens, J. (1986). Bridging the gap between the concrete and the abstract.  Arithmetic Teacher, 33: 14–17.

Hiebert, J., Wearne, D., & Taber, S. (1991). Fourth graders’ gradual construction of

decimal fractions during instruction using different physical representations. The Elementary School Journal, 91: 321–341 

Johnson, L. (2005). Math manipulatives. International Children’s Education. Retrieved Spring  2012 from http://www.iched.org/cms/scripts/page.php?site_id=iched&item_id=math_manipulatives

Klahr, D., Triona, L., & Williams, C. (2007). Hands on what? The relative effectiveness of physical versus virtual materials in an engineering design project by middle school children. Journal of Research in Science Teaching, 44(1), 183-203.

Marsh, L., & Cooke, N. (1996). The Effects of Using Manipulatives in Teaching Math

Problem Solving to Students with Learning Disabilities. Learning Disabilities Research and Practice, 11: 58–65.

Miller, S., & Mercer, C. (1993). Using data to learn about concrete-semi-concrete-abstract instruction for students with math disabilities. Learning Disabilities Research and Practice, 8: 89–96.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.

National Education Association (NEA). (2002). NEA instructional materials survey: Report of findings. Retrieved from http://www.publishers.org/press/pdf/2002%20Instructional%20Materials%20Report.pdf 

Parham, J. (1983). A Meta-Analysis of the use of manipulative materials and student achievement in elementary school mathematics. Dissertation Abstracts International, 96, 44A.

Piaget, J. (1952). The child’s concept of number. New York: Humanities Press.

Piaget, J., & Inhelder, B. (1964). The early growth of logic in the child. New York: W.W. Norton & Company.

Prigge, G. (1978). The differential effects of the use of manipulative aids on the learning of geometric concepts by elementary school children. Journal for Research in Mathematics Education, 9 (5), 361-367.

Raphael, D., & Wahlstrom, M. (1989). The influence of instructional aids on mathematics achievement. Journal for Research in Mathematics Education, 20: 173–190.

Ruzic, R., & O’Connell, K. (2001a). Manipulatives enhancement literature review http://www.cast.org/ncac/Manipulatives1666.cfm .

Ruzic, R., & O’Connell, K. (2001b). Manipulatives. National Center on Accessing the General Curriculum. Retrieved Spring 2006 from http://www.cast.org/ncac/index.cfm?i=1666

Saettler, P. (1990). The evolution of American educational technology. Englewood, CO: Libraries Unlimited, Inc.

Scott, L., & Neufeld, H. (1976). Concrete instruction in elementary school mathematics: Pictorial vs. manipulative. School Science and Mathematics, 76 (1).

Seefeldt, C., & Wasik, B. (2006). Early education: three-, four-, and five-year-olds go to School (2nd ed.). Upper Saddle River, NJ: Pearson Education.

Sowell, E. (1989). Effects of manipulative materials in mathematics instruction. Journal for Research in Mathematics Education, 20: 498–505.

Spear-Swerling, L. (2006, March). The use of manipulatives in mathematics instruction. LD Online. Retrieved Summer 2012 from http://www.ldonline.org/spearswerling/The_Use_of_Manipulatives_in_Mathematics_Instruction

Sutton, J., & Krueger, A. (Eds). (2002). EDThoughts: What we know about mathematics teaching and learning. Aurora, CO: Mid-Continent Research for Education and Learning. 

Suydam, M., & Higgins, J. (1977). Activity-based learning in elementary school mathematics: Recommendations from research. Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

Thornton, C., & Wilmot, B. (1986). Special learners. Arithmetic Teacher, 33: 38–41.

Ward, F. (1971). Montessori method and the American school. Manchester, NH: Ayer Company Publishers.

Wearne, D., & Hiebert, J. (1988). A cognitive approach to meaningful mathematics instruction: Testing a local theory using decimal numbers.” Journal for Research in Mathematics Education, 19(5): 371-384.

 

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