Manipulatives are physical objects used in classrooms to help children learn math.1 Common examples include counters, base-10 blocks, and fraction circle pieces. Many teachers use physical manipulatives along with drawings as useful tools to help students visualize and make sense of fractions.
Teachers may want to expand their fractions toolkit by incorporating virtual manipulatives. Evidence indicates that screen-based versions of physical manipulatives2 offer several advantages that may support students in learning fractions.3
In Illinois, REL Midwest is partnering with education leaders and school districts to develop a Teaching Fractions Toolkit with the aim of narrowing gaps in middle school students' math achievement. This toolkit will support grade 6 teachers in implementing evidence-based practices, including the use of virtual manipulatives, for teaching fractions. The toolkit builds on the What Works Clearinghouse (WWC) practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade.
Below, we highlight the benefits of using virtual manipulatives for teaching fractions and share how they support the partnership's strategic work.
Teachers often use manipulatives and drawings as part of instructional sequences that apply the Concrete Representational Abstract (CRA) framework.4 In this framework, students work with manipulatives as concrete tools for initial exploration of a concept. Students then move into the representational phase and draw pictures to show their thinking. Finally, students connect the work they did with the manipulatives and drawings to abstract numbers and symbols.
When teaching fractions, incorporating manipulatives and drawings into instruction is especially important for correcting common misconceptions that students may have.5 For example, students often reason that 1/5 is greater than 1/2, because 5 is greater than 2. This reasoning makes intuitive sense based on students' experiences with whole numbers, but visual representations of 1/2 and 1/5, as shown in Figure 1, clearly illustrate why 1/2 is the greater value.
Figure 1: Using visual representations to compare 1/2 and 1/5
For the visual representation to be helpful in supporting students' reasoning and correcting misconceptions, however, it has to correctly represent the relative size of the fractions. Many students find it challenging to create accurate drawings of fractions while they are still learning the concepts.6 Students can use physical or virtual manipulatives to create representations like this one before they begin drawing their own representations.
Virtual manipulatives offer several advantages that make them a powerful tool for supporting students in learning about fractions. Virtual manipulatives enable students to interact with digital shapes on a screen. These shapes often resemble familiar physical manipulatives and can be used in similar ways. For example, some virtual fraction circles allow students to put same-size pieces together to represent a fraction sum.
Research studies have found that students who used virtual manipulatives in learning fractions had similar or stronger gains on tests of fraction understanding compared to students who used physical manipulatives.7 Detailed case studies of students using virtual manipulatives to study fraction equivalence and addition also found that the virtual shapes enabled students to discover connections between symbolic and visual representations of fractions and avoid making common errors.8
Examples of the advantage of virtual manipulatives in supporting students as they learn fractions include the following:
Combine features of physical manipulatives and drawings. Virtual manipulatives include useful features of both physical manipulatives and drawings.9 Because they are screen-based, virtual manipulatives have the appearance of a drawing or diagram, yet students can move and rearrange them as with physical manipulatives. Students therefore can work in the concrete and representational phases of the CRA framework at the same time.
For example, when using virtual manipulatives, students and teachers can move models freely in relation to one another (as with physical manipulatives) and simultaneously annotate the models (as with drawings). To create the representation in Figure 2, a learner might create the two circles separately, align and rotate them to make the comparison easier, and then annotate to more clearly illustrate that the yellow shading in the circle on the right takes up more space than the yellow shading in the circle on the left. Then the learner could write their comparison symbolically: 1/4 < 3/10.
Figure 2: Aligning and annotating virtual manipulatives
The ability to move models freely makes it easy for students to align or overlap visual models to compare the size of fractions. The ability to annotate visual models can help students make connections between representations10 and connect visual models to real-world contexts (as discussed in Recommendations 1 and 3 in the WWC practice guide Developing Effective Fractions Instruction in Kindergarten Through 8th Grade). With virtual manipulatives, teachers and students have access to both of these powerful and helpful features at the same time.
Provide increased flexibility, speed, and accessibility. Because virtual manipulatives are digital and interactive, they offer several unique features that are difficult to replicate with physical tools and hand drawings. For example, some virtual manipulatives allow students to instantly partition a shape into equal-sized parts. Physical fraction manipulatives are often restricted to a small set of common denominators—such as halves, thirds, fourths, eighths, and tenths. On the other hand, virtual manipulatives enable students to create visuals of fractions with less common denominators, such as sevenths, or with large denominators, such as 100ths. And students can create and accurately partition shapes with far less time and effort than with drawings on paper.
The ability of virtual manipulatives to quickly create precise representations of a range of fractions can support learning by
For example, students may be more likely to notice patterns in the numerators and denominators of equivalent fractions when they are able to generate more examples of equivalent fractions in a class period. As described in Recommendation 3 of the WWC practice guide Developing Effective Fractions Instruction for Kindergarten Through 8th Grade, students' ability to note such patterns and explore them with more examples is an important step in understanding why fraction procedures, such as the multiplication rule for generating equivalent fractions, make sense and produce correct answers.
Across the coming years, REL Midwest will continue to partner with grade 6 math teachers, math leaders, and administrators in Illinois school districts to develop, test, and refine the Teaching Fractions Toolkit. This toolkit will consist of two components:
Browse the following resources to learn more about the Teaching Fractions Toolkit and evidence-based practices for teaching fractions.
1 Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380–400. https://eric.ed.gov/?q=EJ1007941
2 Moyer-Packenham, P. S., & Bolyard, J. J. (2016). Revisiting the definition of a virtual manipulative. In P. S. Moyer-Packenham (Ed.), International Perspectives on Teaching and Learning Mathematics with Virtual Manipulatives (pp. 3–23). Springer International Publishing.
3 Moyer-Packenham, P. S., & Westenskow, A. (2013). Effects of virtual manipulatives on student achievement and mathematics learning. International Journal of Virtual and Personal Learning Environments, 4(3), 35–50. https://eric.ed.gov/?q=EJ1154970&id=EJ1154970
4 Agrawal, J., & Morin, L. L. (2016). Evidence-based practices: Applications of concrete representational abstract framework across math concepts for students with math disabilities. Learning Disabilities Research and Practice, 31(1), 34–44. https://eric.ed.gov/?q=EJ1089034&id=EJ1089034
5 Stafylidou, S., & Vosniadou, S. (2004). The development of students' understanding of the numerical value of fractions. Learning and Instruction, 14(5 SPEC.ISS.), 503–518. https://eric.ed.gov/?q=EJ731639&id=EJ731639
6 Bouck, E. C., Bassette, L., Shurr, J., Park, J., Kerr, J., & Whorley, A. (2017). Teaching equivalent fractions to secondary students with disabilities via the virtual-representational-abstract instructional sequence. Journal of Special Education Technology, 32(4), 220–231. https://eric.ed.gov/?q=EJ1159688&id=EJ1159688
7 Moyer-Packenham, P. S., & Suh, J. M. (2012). Learning mathematics with technology: The influence of virtual manipulatives on different achievement groups. Journal of Computers in Mathematics and Science Teaching, 31(1), 39–59. https://eric.ed.gov/?q=EJ973330&id=EJ973330; Moyer-Packenham & Westenskow, 2013.
8 Reimer, K., & Moyer, P. S. (2005). Third-graders learn about fractions using virtual manipulatives: A classroom study. The Journal of Computers in Mathematics and Science Teaching, 24(1), 5–25. https://eric.ed.gov/?q=EJ724762&id=EJ724762; Suh, J., Moyer, P. S., & Heo, H.-J. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4). https://eric.ed.gov/?q=EJ1066648&id=EJ1066648
9 Moyer-Packenham, P. S., Salkind, G., & Bolyard, J. J. (2008). Virtual manipulatives used by K-8 teachers for mathematics instruction: Considering mathematical, cognitive, and pedagogical fidelity. Contemporary Issues in Technology and Teacher Education, 8(3), 202–218. https://eric.ed.gov/?q=EJ1080563&id=EJ1080563
10 Rau, M. A., & Matthews, P. G. (2017). How to make "more" better? Principles for effective use of multiple representations to enhance students' learning about fractions. ZDM: The International Journal on Mathematics Education, 49(4), 531–544. https://eric.ed.gov/?q=EJ1149156&id=EJ1149156
11 Bouck et al., 2017.
12 Suh, J. M. (2010). Leveraging cognitive technology tools to expand opportunities for critical thinking in elementary mathematics. Journal of Computers in Mathematics and Science Teaching, 29(3), 289–302. https://eric.ed.gov/?q=EJ896827&id=EJ896827
13 Moyer-Packenham & Suh, 2012.