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The Math Manipulatives Hiding in a Junk Drawer

When parents ask how to get the benefits of manipulatives like base 10 blocks in distance learning, teachers can share strategies for using common household items.

Math manipulatives, as teachers know, help students develop conceptual understanding by empowering them to build concrete models of abstract ideas. The play that manipulatives encourage is done with purpose, not at random. It provides a common language with which students can describe phenomena and communicate with peers and teachers.

With manipulatives, students create a multidimensional framework on which their math foundation can grow and abstract ideas can come to life. Evidence-based research pointing to the efficacy of the use of manipulatives is strong and shows that student achievement improves over the long term  when they are used.

If manipulatives are a key component of your hands-on instructional strategy and these days you’re teaching remotely, you likely have parents with questions about what they can use to replace classroom supplies like color cubes and tiles, bug counters, and base 10 blocks.

The answer is easy: household items found in pantries, toy boxes, sewing kits, and junk drawers.

When parents ask what objects they can use in place of classroom manipulatives, first review with them the concept being taught and explain why the manipulative is used—how it fits in with a specific learning approach. Then share possible substitutions.

With counters, for example, explain to the parent how classroom versions (plastic animals or round counters) are used to help students learn quantity or do addition and subtraction; then suggest alternate items that are alike in size, shape, and possibly color. Dried beans, buttons, and coins are easy substitutes. Those same manipulatives can also work for modeling multiplication or division by combining or creating groups.

For fractions, explain to parents how students use manipulatives such as the counters to arrange items, often of different colors, in groups and identify a portion of a group. Then suggest buttons or beans of different colors. Paper strips cut to equal length and then folded, one in thirds, another in quarters, and so on, can also help students understand which fraction is larger. (Note that the tactile aspect of physically folding the paper is more concrete than a representation by drawing and can really make a difference in the learning component.)

Math Manipulatives From Household Items

Here are additional ways you can offer parents support as they search for manipulatives that can be found or created in the home:

10 frames: First explain what a 10 frame is (a 2x5 rectangular frame of squares) and how they support instruction (they give students visual benchmarks to help see how numbers fit together and develop number sense; teachers often have magnetic versions).

Share with parents how to draw a 10 frame at home: two adjacent, parallel rows of five squares each on a sheet of paper.

To go 3D with 10 frames, suggest that the parent use an egg carton: Cut off the lid and one of the ends, leaving spots for what would have been 10 rather than 12 eggs.

For either version, advise parents to find small objects that fit in the spaces to count.

Linking cubes:  Explain to parents how these interlocking, different-colored plastic blocks are great for topics from counting to bar graphs to fractions, and that some varieties of linking cubes connect end to end to make towers, while others snap together on all sides to form geometric shapes. You can suggest components of children’s building-block toys that are similar and can be substituted, or you can suggest using different-colored squares of paper.

Base 10 blocks:  Explain to parents how these help children understand 10 and see how place value impacts numbers; students use them to build number combinations, practice regrouping, and discover how they can make “trades” to build larger pieces or break down a figure into smaller pieces. Explain how regrouping is the beginning of understanding the process of carrying or borrowing.

A few suggestions for home substitutes include linking cubes (interlocking plastic blocks) or craft sticks, which students can bundle into groups of 10 with rubber bands. Cutting pieces from graph paper or using pipe cleaners strung with 10 beads for a 10s place would work, too.

Geoboards:  Explain to parents how these boards with uniformly spaced, elevated pegs and rubber bands are used to help students understand shapes, areas (arrays), and graphing.

At home, a pegboard or cork board with a uniformly spaced grid will do. Parents can supply pegs or thumbtacks and rubber bands or string.

This is just a start; other teachers and parents will have additional ideas. Keep in mind that virtual models of these manipulatives are available online and offer another mode for students using the same approach.

Manipulatives in Mathematics Education

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• Maria Giuseppina Bartolini 2 &
• Francesca Martignone 2  

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Bartolini, M.G., Martignone, F. (2020). Manipulatives in Mathematics Education. In: Lerman, S. (eds) Encyclopedia of Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-030-15789-0_93

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In a traditional grade-level mathematics classroom, the use of manipulatives has become essential in providing students with the knowledge to conceptualize basic math operation skills. This approach to instruction involves using physical tools to enhance student understanding of the mathematical content. Teachers are finding the need for using manipulatives to create effective, active, and engaging math lessons. Using manipulatives, or “ tangible objects,” can provide for a variety of learning styles and abilities within classrooms (Horan & Carr, 2018). 

What are Manipulatives?

Horan and Carr (2018) define manipulatives as concrete objects that allow students hands-on experience while being actively engaged in the learning. There are multiple ways to use manipulatives. In the classroom, teachers are using manipulatives in a lesson as they introduce, practice or remediate a mathematical concept (Hidayah et al., 2021). These physical tools may include a variety of concrete objects that might be used at the elementary level such as counters, fraction strips, pattern blocks, cubes, geoboards, etc., for all kinds of math instruction. 

Using manipulatives as an approach provides a foundation which will encourage critical thinking and students' ownership of their work. Teachers are able to have a vivid picture of student understanding in which they can determine the next appropriate steps (McDonough, 2016). 

Origin of Manipulatives

Using math manipulatives dates back to earlier civilizations that used clay beads and wooden trays to help grasp mathematical concepts (Boggan et al., 2010). Throughout history, different types of manipulatives have been used to aid in comprehension of mathematical concepts. 

We first hear about manipulatives being seen as educational tools for teaching in the late 1800s. Teachers were starting to use manipulatives to enhance their lessons and saw positive outcomes in their students' mathematical skills. In the 1900s, Italian physician and educator Maria Montessori developed the use of manipulatives with the goal in mind to enable children to learn through personal investigation and exploration (Hurst &  Linsell, 2020). Today, using manipulatives stresses the importance of concrete operations in the primary stages of knowledge formation in young children. In a traditional mathematics class today, using manipulatives is well-established in the classroom. 

Why are manipulatives important?

Based on psychologist Jean Piaget’s research, children learn concepts through three levels of knowledge: concrete, pictorial, and abstract (Hurst &  Linsell, 2020). As students manipulate objects, they take the necessary first steps toward building understanding and internalizing math processes and procedures. Manipulating objects allows students to explore concepts at the first, or concrete level of understanding. Strategies and algorithms will be developed over time (Ojose, 2008).

Students need to understand the concept at the two levels of concrete and pictorial first before they can handle an abstract or symbolic level (Hurst &  Linsell, 2020). To create mental images and models, it is necessary to use concrete manipulatives. Students who show an understanding of the concept at this physical or concrete level are well-positioned to move to the next level where they will be able to use representations of the objects in place of the real objects (Tirosh et al., 2018). 

The use of concrete models can facilitate the development of number sense as well as develop the meaning of written symbols and help students develop a sense of place value (Hurst &  Linsell, 2020). By using this method, teachers can get a better understanding of what students know, as well as identify misconceptions, so they can design interventions accordingly.

Understanding the interconnections of mathematical ideas can be improved by utilizing manipulatives. Using manipulatives to solve a problem can assist students in keeping track of what they did and explaining their ideas (Hurst &  Linsell, 2020).

Student-Centered Approach

Student-centered learning has a variety of meanings in education. Students are encouraged to engage with their own ideas, experiment with new materials, and explore. A common description of student-centered learning is that students are at the center of their learning where the teacher is there to support and guide students’ progress and learning (Keiler, 2018). So what makes math manipulatives student-centered? 

Using math manipulatives fosters student engagement in a way that allows for students to explore different math concepts with hands-on learning materials (Hidayah et al., 2021). In an encyclopedia article, Stephan (2014) stated the following: 

Using math manipulatives is a learner-centered teaching approach to mathematics instruction that places heavy emphasis on the students taking responsibility for problem solving and inquiry. The teacher is viewed as a facilitator by posing problems and guiding students as they work with partners toward creating a solution. (p. 331) 

Therefore, manipulatives are student-centered because students are able to play with tangible objects which are designed to give students a deeper understanding of mathematical concepts. 

Outcomes of using Math Manipulatives

The use of manipulatives in the classroom greatly aids the development of strong mathematical foundations in young students. Research shows that there are benefits to using manipulatives to help teach a mathematical concept. 

Academic 

According to D’angelo and Iliev (2012), using manipulatives aids in furthering student comprehension of mathematical knowledge. As students are given the chance to explore on their own with the chosen manipulatives they are able to critically think and make connections in understanding the math concept. Data have shown that concrete objects can help children gain access to concepts and processes that might otherwise be inaccessible (Uttal, 1997). Looking at a specific group of students, English language learners’ (ELLs) comprehension increases immensely. Data have shown that ELLs, “improve in vocabulary development, oral proficiency, comprehension, and display enthusiasm to continue using the manipulatives” (Stapleton, 2014, p. 161). ELL students' comprehension increased because they had to interpret a directive with an action in solving the problem. Therefore, the use of hands-on, multi-sensory manipulatives to help students increase comprehension is encouraged. 

Another connection is how the role of manipulatives and metacognition go hand in hand with young children's cognitive development. Metacognition is when one observes, tries, and reasons with various mathematical concepts. It is thinking about thinking; a way for student learning to be enhanced and for them to understand their own learning processes. Belenky et al. (2009) state, “metacognitive prompts are questions that ask students to reflect on various aspects of the learning materials and problem-solving process and have been hypothesized to facilitate abstraction and learning” (p. 103). Students given concrete manipulatives with metacognitive prompts have shown a better transfer of procedural skills than students given abstract manipulatives with problem focused prompts. As a result, the manipulatives utilized in mastering sophisticated cognitive skills taught in mathematics are critical to increasing comprehension.

The use of multi-sensory manipulatives as tools has been said to increase involvement and interaction in teaching ESL students. In a journal article, Stapleton (2014) stated the following: 

Students enjoy working with hands-on manipulatives which increase the opportunity for student involvement and interaction. Students who use the materials do not sit passively while the instructor attempts to verbally explain a concept. Students are encouraged to participate with other students, make connections with new concepts, and draw conclusions based on their understanding. (p. 162) 

This brings us to the next point: visualization. Where some students learn best with visuals, math manipulatives also aid with being able to conceptualize a math problem (Carbonneau, 2013). While students can recall material from books and lectures for short periods of time, deep understanding and the ability to apply what they've learned to new contexts necessitate conceptual understanding anchored in actual interactions with concrete objects (D’angelo & Iliev, 2012). 

Research shows that when manipulatives in mathematics are used effectively, student understanding and engagement increases because manipulatives aid in the understanding of visual concepts through the use of visuals, scaffolding learning, and engaging students in learning (Cockett, 2015). Students are able to link representations based on manipulatives with written, symbolic representations. 

Affective 

Authors Cockett and Kilgour (2015) did a quantitative study on the impact of using manipulatives in mathematics on student understanding, efficiency, engagement and enjoyment. During this study, several types of manipulatives were used with students participating in various mathematical activities. Observations were also part of collecting qualitative data. The results concluded that students were more engaged when using manipulatives, and that their perception of their learning environment improved in each of the three areas: enjoyment, understanding, and efficiency. 

In addition to enjoyment, concrete things that imitate daily objects help youngsters learn concepts by allowing them to draw on their practical expertise. Students are building up their problem solving skills and making connections. Planning instructional engagement activities is a huge part of students' motivation. Manipulatives give that extra boost in creativity and an increase in skills in students. A Yale University study (Hurst & Linsell, 2020) found that simple objects kept elementary students involved and entertained with very high levels of attention and concentration. Manipulatives also allowed students to design and experiment to find a solution, which encourages social interaction (Berk, 1999). 

Therefore, manipulatives are effective for the following reasons: they are multisensory, they represent ideas in more than one way, they promote communication among students, and they increase confidence, leading to less confusion and a deeper understanding. 

Challenges with Manipulatives 

Challenges are a natural part of mathematics. Research has confirmed that using math manipulatives produces positive outcomes in students’ cognitive development and skills; however, there are some challenges with using them. When students learn with manipulatives, they may become too reliant on the item and context (Boggan et al., 2010). If students are constantly using manipulatives, they might become a crutch, preventing students from learning more advanced problem-solving skills (Boggan et al., 2010). Students will have difficulty transferring new knowledge to new contexts (Boggan et al., 2010). 

Effectiveness of Learning 

Hidayah et al., (2021) stated, “the use of manipulatives is still limited to the use of classical and group learning. The students, therefore, could not repeat the math manipulatives instruction by themselves after class” (p. 539). The manipulatives' nature allows students to manipulate them in order to learn certain ideas. It is necessary to have manipulatives, but it is also important to know how to utilize them appropriately in a well-designed learning experience.

Manipulatives, like any other educational instrument, may aid or impede learning. 

Using manipulatives is of value in the mathematics classroom, especially when students are making their own connections to problem-solving in relation to mathematical concepts. 

When teaching mathematics, educators who are aware of their students' competency levels can effectively scaffold content. To do so, teachers must first comprehend how their students think and why they think that way. Mathematical knowledge acquisition in early learners is dependent on student-centered mathematics education; consequently, educators should endeavor to provide a mathematically rich atmosphere in which children critically explore concepts, solve problems, and openly discuss their thoughts. Teachers who use tangible manipulatives effectively in their classrooms can have a favorable impact on their students' arithmetic skills. When it comes to employing manipulatives in the classroom, the advantages are infinite. The use of these tools enhances students' learning experiences, bridges the gap between the physical and abstract, and, ultimately, fosters life-long learning in curious young learners.

Belenky, D. M., & Nokes, T. J. (2009) Examining the role of manipulatives and metacognition on engagement, learning, and transfer. The Journal of Problem Solving,   2 (2), 6. DOI: 10.7771/1932-6246.1061

Berk, E. G. (1999). Hands-on science: Using manipulatives in the classroom. Principal , 78 (4), 52.

Boggan, M., Harper, S., & Whitmire, A. (2010). Using manipulatives to teach elementary mathematics. Journal of Instructional Pedagogies, 3 (1), 1-6. 

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.

Cockett, A., & Kilgour, P. W. (2015). Mathematical manipulatives: Creating an environment for understanding, efficiency, engagement, and enjoyment. Teach Collection of Christian Education , 1 (1), 5.

D'angelo, F., & Iliev, N. (2012). Teaching mathematics to young children through the use of concrete and virtual manipulatives (ED534228). ERIC. https://files.eric.ed.gov/fulltext/ED534228.pdf

Hidayah, I., Isnarto, Masrukan, Asikin, M., & Margunani. (2021). Quality management of mathematics manipulative products to support students’ higher order thinking Skills. International Journal of Instruction , 14 (1), 537–554.

Horan, E., & Carr, M. (2018). How much guidance do students need? An intervention study on kindergarten mathematics with manipulatives. International Journal of Educational Psychology , 7 (3), 286–316.

Hurst, C., & Linsell, C. (2020). Manipulatives and multiplicative thinking. European Journal of STEM Education , 5 (1), 04. 

Keiler, L.S. Teachers’ roles and identities in student-centered classrooms. IJ STEM Ed, 5 , 34 (2018). https://edtechbooks.org/-TrXd

McDonough, A. (2016). Good concrete activity is good mental activity. Australian Primary Mathematics Classroom, 21 (1), 3–7.

Ojose, B. (2008). Applying Piaget's theory of cognitive development to mathematics instruction. The Mathematics Educator , 18 (1), 26-30. 

Stapleton, T. J. (2014). Multi-sensory, hands-on manipulatives and adult ESL. NAMTA Journal , 39 (3), 153-169.

Stephan M. (2014) Learner-centered teaching in mathematics education. In: Lerman S. (eds) Encyclopedia of Mathematics Education. Springer, Dordrecht. https://edtechbooks.org/-wVNq  

Tirosh, D., Tsamir, P., Barkai, R., & Levenson, E. (2018). Engaging young children with mathematical activities involving different representations: Triangles, patterns, and counting objects. Center for Educational Policy Studies Journal , 8 (2), 9–30.

Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology , 18 (1), 37-54.

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Math Manipulatives: A Student-Centered Approach to Teaching Mathematics

by Emily Nash

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Posted in: Aha! Blog > Eureka Math Blog > Manipulatives Instructional Design Eureka Math Squared > Math Manipulatives: A Student-Centered Approach to Teaching Mathematics

In a comprehensive review of mathematics education research, the National Research Council (2001) concluded “the evidence indicates, in short, that manipulatives can provide valuable support for student learning when teachers interact over time with the students to help them build links between the object, the symbol, and the mathematical idea both represent.”

Meaningful Built-In Tools for Deeper Learning

Manipulatives have long been an essential part of mathematics instructional programs . Why? Because r esearch shows that the consistent and appropriate use of manipulatives increases student engagement and understanding of key mathematics concepts (Dinsmoor 2022 ; Hand2Mind n.d. ) . Manipulatives increase student engagement and provide an access point to abstract concepts by serving as a concrete model of the math concept. This model can be called upon later as students continue to practice and develop the skills connected to this learning . When students represent a math concept with a physical manipulative, they better understand the concept. 

Research also shows that concrete objects can help children access math concepts and processes that might otherwise be inaccessible to them (Dinsmoor 2022; Hand2Mind n.d.) . For example, Gersten et al. (2009) summarized research on the use of concrete objects—both formal and informal manipulatives—with struggling math students. In most studies, the effect size of student math learning with manipulative use was significantly larger than when manipulatives were not used.  

Drawing upon the research, and recognizing the impact that manipulatives can have on learning, the Eureka Math 2 ® writers and mathematicians have   

• integrated the use of manipulatives throughout all grades levels as part of a concrete-pictorial-symbolic (CPS) progression,
• planned for coherent models to be used as students move up the grade levels, and
• created unique manipulative items, when needed. 

Strategically S caffolded L earning

To develop the conceptual understanding needed for future success, students need access to a wide array of tools that help them grasp , inte grate , and internalize mathematical concepts. Multiple representations of concepts , when blended into a deliberate sequence—like the concrete - pictorial - symbolic (CPS) progression recommended by the research —are an effective way to lead students to a richer understanding of abstract concepts (Dinsmoor 2022; Hand2Mind n.d.) . The CPS approach is a critical instructional design element of Eureka Math 2 that facilitates deep learning and works best when strong attention is directed at developing student understanding in the concrete and pictorial stages. Here is an example of a concrete- pictorial -symbolic progression in Eureka Math 2 :  

Purposeful exploration of math concepts using tangible, hands-on objects like pattern blocks, tiles, and cubes is a vital first step to building conceptual understanding. As students become more comfortable with the concepts and skills being taught, they transition toward visualizing and drawing the concept using pictures. Then, in a continued and intentional progression that occurs within and across modules and levels, students demonstrate their understanding of the concept using the symbolic notation of numerals and symbols.    

"Mathematically proficient students consider the available tools when solving a mathematical problem…Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.” (Illustrative Mathematics 2014)

Eureka Math 2 is unique because it intentionally builds bridges between the CPS stages through modeling and guided practice. Students feel empowered to use more abstract representations of concepts because they understand how the mathematical model is connected to the concept and how the concrete, pictorial, and symbolic representations are connected to one another. The curriculum also overlaps the stages to help students build familiarity with the next stage in the progression, even when they are primarily focused on an earlier one. Finally, the CPS instructional approach in Eureka Math 2 offers explicit scaffolding opportunities, enabling teachers to effectively move students back to earlier stages as needed so students can successfully access more advanced learning.  

T his exposure to various mathematical tools , along with intentional guidance and scaffolding , ultimately develop s students’ capacity to “ U se appropriate tools strategically,” one of the mathematical habits of mind . In a Eureka Math 2 classroom, students gain an understanding of why and how manipulatives and other tools aid their problem-solving processes , and students are empowered to select the appropriate tools for their learning and application of mathematical concepts.

A Coherent Progression of Models

Manipulatives can be a part of a coherent set of concrete representations that students can draw on throughout grade levels. These concrete representations help build background knowledge in a way that activates students’ memory and emphasizes how the same math concepts can apply to new, more complex units. Many models used in Grade Levels K–5 evolve along with the growing complexities of mathematics and are representative of similar models in Grade Level 6–Algebra I. For example:  

• The number path is an early linear model representing whole numbers. This model evolves into using number lines to represent a more extensive variety of number types, including fractions and decimals.  
• Place value disks and place value charts are used starting in Grade Level 2 and through Grade Level 5 to model numbers and to compute with understanding.
• Centimeter cubes are used across the levels, starting in Grade Level K. 

Thoughtfully Developed Tools  

Our teacher–writers and mathematicians embedded engaging, hands-on, concrete models into the Eureka Math 2 curriculum. When the right tool to represent the concept didn’t exist, they created it.   

In Grade Level 2 Module 1, students are introduced to meter sticks and measuring tapes as tools to explore metric measurement and the beginning concepts of place value. The Eureka Math 2 exclusive Double-Sided Meter Sticks and Measuring Tapes are specifically designed to be used as a concrete model throughout Grade Level 2.

Many of our unique manipulatives, such as Story Boards, Block Puzzles, and Bingo Games can be utilized for games and centers throughout the school year.  

When you purchase the manipulatives kits for your grade level, you utilize the tools exactly as the expert teacher–writers intended. In addition, h a v i n g th ese vital tools a t the p o i n t o f u s e means teachers can spend less time prepping a n d m o r e t i m e i n t e n t i o n a l l y p l a n n i n g. Take a look inside the manipulatives kits for Level 2 of Eureka Math 2 . 

Built-In Support for Educators

Manipulatives kits are available for each grade level of Eureka Math 2 . There are also virtual versions of many (but not all) of the manipulatives that can be displayed to the class. To ease instructional planning and delivery, all the materials needed for instruction can always be found in the Module Overview for each module on the Great Minds ® Digital Platform. 

In the Teach book, materials can be found listed at the end of each module.

Since teachers are key to helping students use manipulatives successfully, the Eureka Math 2 writers also share insights into the design of certain manipulative activities in the “Why” section of each module to support teacher understanding. You can see an example of this feature below from the Grade Level 2 Module 1 Module Overview. 

L earn more about how Eureka Math 2 uses manipulatives to build math knowledge .  

Works Cited

Dinsmoor, K. 2022. “Math Manipulatives .” In S. L. Mason (Ed.), Student-Centered Approaches in K–12 and Higher Education. EdTech Books. https://edtechbooks.org/student_centered/math_manipulatives. 

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. 2009. “Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060).” Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies. ed.gov/ncee/wwc/publications/practiceguides/. 

Hand2Mind. “Research on the Benefits of Manipulatives.” Accessed February 21, 2023. https://www.hand2mind.com/media/contentmanager/content/Benefits_of_Manipulatives.pdf. 

Illustrative Mathematics. 2014. “Standards for Mathematical Practice: Commentary and Elaborations for K–5.” Tucson, AZ. https://commoncoretools.me/wp-content/uploads/2014/02/Elaborations.pdf. 

National Research Council. 2001. “Adding It Up: Helping Children Learn Mathematics.” Washington, DC: The National Academies Press. https://doi.org/10.17226/9822. 

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Emily Nash is the associate product marketing manager for Eureka Math and Eureka Math². She previously taught middle school mathematics and is excited to share those experiences as part of the Great Minds team.

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Accommodations Toolkit

Manipulatives: research.

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This fact sheet on manipulatives is part of the Accommodations Toolkit published by the National Center on Educational Outcomes (NCEO). It summarizes information and research findings on manipulatives as an accommodation. This toolkit also contains a summary of states’ accessibility policies for manipulatives .

  What are manipulatives? Manipulatives can be either physical or virtual and are typically used to help with mathematical calculations.

• “Physical manipulatives (PMs) are 3D physical (real world) objects that students can touch” with their hands (Ha & Fang, 2018). Often, they are some forms of blocks used for counting.
• “Virtual manipulatives (VMs) are 3D computer graphics (virtual world) that students can see and manipulate on a computer screen” (Ha & Fang, 2018).
• There are also interactive virtual and physical manipulatives (VPMs) that combine the two types of manipulatives in which students can observe virtual manipulative in real-time while they handle physical manipulatives.

What are the research findings on who should use this accommodation? Research showed that the students who benefited most from using manipulatives often were students with learning disabilities (LD), autism spectrum disorder (ASD), or mild intellectual disabilities (Bassette et al., 2019; Bouck et al., 2018, 2020). 

What are the research findings on implementation of manipulatives? Seven studies were located that addressed the effects of manipulatives use on student mathematics assessment outcomes. 

• Three studies compared virtual and physical manipulatives with mixed findings. Bassette et al. (2020) found that some students with ASD were more efficient with app-based (virtual) manipulatives while others were more efficient with concrete (physical) manipulatives. However, another study found no difference in performance between an app-based and a hand-held manipulative (Bouck et al., 2018). The third study found that virtual and physical manipulatives were helpful for students with intellectual and developmental disabilities and improved independence in two-step addition and subtraction problems. However, the students struggled to generalize strategies across problems without using manipulatives (Long, et al., 2020).
• Four studies explored the effectiveness of virtual manipulatives and found that they improved student performance or increased speed. One study found that students with ASD completed more steps independently per minute during subtraction problems with help from an app-based manipulative than without the use of a manipulative (Bassette et al., 2019). Another study found that virtual manipulatives helped improve accuracy on long division problems for middle school students with LD (Bouck et al., 2020), while still another study found that virtual manipulatives were beneficial for teaching higher-order mathematical concepts to secondary students with LD (Satsangi, Hammer, & Hogan, 2018). Lastly, Satsangi, Hammer, and Evmenova (2018) found higher performance for students with LD who were solving multistep linear equations using virtual manipulatives than without. 

What perceptions do students and teachers have about manipulatives? Five studies touched on teacher or student perceptions regarding the use manipulatives when taking an assessment. 

• Two studies examined teacher perceptions of manipulatives. The findings were mixed regarding whether the teachers believed manipulatives were useful during assessment. Tindal et al. (2008) found that teachers felt strongly that the use of manipulatives during testing was potentially beneficial. Another study found that almost all teachers believed that physical manipulatives helped students learn early numeracy concepts, but nearly half of these teachers indicated that students had difficulty using them during testing unless they had previously been used during math instruction (Jimenez & Stanger, 2017).
• Two studies examined the perceptions of students with LD regarding the usefulness of manipulatives. Both studies found that students considered them helpful. Ha and Fang (2018) found that students thought both virtual manipulatives and physical manipulatives were helpful. However, they preferred to use both virtual and physical manipulatives together instead of just using one or the other. Another study on virtual manipulatives found that many students with LD found virtual manipulatives helpful when solving math problems (Satsangi, Hammer, & Hogan, 2018).
• One study examined whether students with ASD preferred virtual or physical manipulatives. This study found that all students preferred using app-based manipulatives over concrete physical manipulatives, even when they performed better with physical manipulatives (Bassette et al., 2019).

  What have we learned overall? The research showed that the use of either physical or virtual manipulatives improved mathematics performance. This accommodation may be especially helpful for students with LD, ASD, and mild intellectual disabilities. Both teachers and students perceived manipulatives of all kinds (i.e., virtual manipulatives, physical manipulatives, combination of virtual and physical manipulatives) as helpful. Virtual manipulatives were preferred by students, though students may be more likely to get the correct answer when using physical manipulatives. There is limited research on the effectiveness of physical manipulations during testing though several recent studies examined virtual manipulatives. All studies examined the use of manipulatives during math assessments; no research was found that examined the effectiveness of manipulatives for other content assessments (e.g., science). There is also a need for research on the use of manipulatives during assessment by students who are blind or have low vision. Students who are blind or have low vision often use manipulatives, but no research was found on the potential usefulness of manipulatives for this group.

Bassette, L., Bouck, E., Shurr, J., & Park, J. (2019). Comparison of concrete and app-based manipulatives to teach subtraction skills to elementary students with autism . Education and Training in Autism and Developmental Disabilities , 54 (4), 391–405. http://www.daddcec.com/etadd.html

Bassette, L., Bouck, E., Shurr, J., Park, J., Cremeans, M., Rork, E., Miller, K., & Geiser, S. (2020). A comparison of manipulative use on mathematics efficiency in elementary students with autism spectrum disorder . Journal of Special Education Technology , 35 (4), 179–190. https://journals.sagepub.com/home/jst

Bouck, E., Park, J., & Stenzel, K. (2020). Virtual manipulatives as assistive technology to support students with disabilities with mathematics . Preventing School Failure: Alternative Education for Children and Youth , 64 (4), 28–289. https://doi.org/https://doi.org/10.1080/1045988x.2020.17621157

Bouck, E., Shurr, J., Bassette, L., Park, J., & Whorley, A. (2018). Adding it up: Comparing concrete and app-based manipulatives to support students with disabilities with adding fractions . Journal of Special Education Technology , 33 (3), 194–206. https://doi.org/10.1177/0162643418759341

Ha, O., & Fang, N. (2018). Interactive virtual and physical manipulatives for improving students’ spatial skills . Journal of Educational Computing Research , 55 (8), 1088–1110. https://doi.org/10.1177/0735633117697730

Jimenez, B. A., & Stanger, C. (2017). Math manipulatives for students with severe intellectual disability: A survey of special education teachers . Physical Disabilities:  Education and Related Services , 36 (1), 1–12. https://doi.org/10.14434/pders.v36i1.22172

Long, H., Bouck, E., & Domka, A. (2020). Manipulating algebra: Comparing concrete and virtual algebra tiles for students with intellectual and developmental disabilities . Exceptionality . https://doi.org/10.1080/09362835.2020.1850454

Sastangi, R., Hammer, R., & Hogan, C. D. (2018). Studying virtual manipulatives paired with explicit instruction to teach algebraic equations to students with learning disabilities . Learning Disabilities Quarterly , 41 (4), 227–242. http://journals.sagepub.com/home/ldq

Satsangi, R., Hammer, R., & Evmenova, A. S. (2018). Teaching multistep equations with virtual manipulatives to secondary students with learning disabilities . Learning Disabilities Research & Practice , 3 (2), 99–111. https://onlinelibrary.wiley.com/journal/15405826

Tindal, G., Lee, D., & Ketterlin-Geller, L. (2008). The reliability of teacher decision-making in recommending accommodations for large-scale tests (Technical Report No. 08-01). Behavioral Research and Teaching, University of Oregon. http://www.brtprojects.org/publications/dl/61

Attribution

All rights reserved. Any or all portions of this document may be reproduced and distributed without prior permission, provided the source is cited as:

Goldstone, L., Hendrickson, K., Lazarus, S., & Fleming, K. (2021). Manipulatives: Research (NCEO Accommodation Toolkit #12a) . National Center on Educational Outcomes.

NCEO is supported through a Cooperative Agreement (#H326G160001) with the Research to Practice Division, Office of Special Education Programs, U.S. Department of Education. The Center is affiliated with the Institute on Community Integration at the College of Education and Human Development, University of Minnesota. NCEO does not endorse any of the commercial products used in the studies. The contents of this report were developed under the Cooperative Agreement from the U.S. Department of Education but do not necessarily represent the policy or opinions of the U.S. Department of Education or Offices within it. Readers should not assume endorsement by the federal government. Project Officer: David Egnor.

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Qualitative Research on Math Manipulatives in Montessori and Traditional Elementary 1st-3rd Grade Classrooms

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