Manipulatives are appropriate for all ages and grade levels. In the past, some educators have mistakenly believed that manipulatives are only effective for younger children or students with special needs, but that isn’t the case. In 2013, the National Council of Supervisors of Mathematics (NCSM) issued a position statement on the use of manipulatives in classroom instruction to improve student achievement. “[I]n order to develop every student’s mathematical proficiency, leaders and teachers must systematically integrate the use of concrete and virtual manipulatives into classroom instruction at all grade levels.” (NCSM, 2013)] This position is based on research supporting the use of manipulatives in classroom instruction.

Manipulatives should be used beyond elementary school. Many middle school students are still in the developmental stage where they are moving into being able to think abstractly, but they aren't there yet. Struggling high school students also benefit from manipulatives because it gives them an anchor to remember the lesson and reduces the stress of thinking abstractly.

Manipulatives should be used frequently in all classrooms. Wenglinsky’s (2000) analysis of NAEP data suggests the value of interaction over time. By examining data about classroom activity, this analysis suggests that, “when students are exposed to hands-on learning on a weekly rather than a monthly basis, they prove to be 72% of a grade level ahead in mathematics."

Manipulatives should also be used in all areas of math. Studies have shown that students using manipulatives in specific mathematical subjects are more likely to achieve success than students who don’t have the opportunity to work with manipulatives. Following are some specific areas in which research shows manipulatives are especially helpful:

Counting - Some children need to use manipulatives to learn to count (Clements, 1999).

Place Value - Using manipulatives increases students’ understanding of place value (Phillips, 1989).

Computation - Students learning computational skills tend to master and retain these skills more fully when manipulatives are used as part of their instruction (Carroll and Porter, 1997).

Problem Solving - Using manipulatives has been shown to help students reduce errors and increase their scores on tests that require them to solve problems (Carroll and Porter, 1997; Clements, 1999; Krach, 1998).

Fractions - Students who have appropriate manipulatives to help them learn fractions outperform students who rely only on textbooks when tested on these concepts (Jordan, Miller, and Mercer, 1998; Sebesta and Martin, 2004).

Ratios - Students who have appropriate manipulatives to help them learn fractions also have significantly improved achievement when tested on ratios when compared to students who do not have exposure to these manipulatives (Jordan, Miller, and Mercer, 1998).

Algebraic Abilities - Algebraic abilities include the ability to represent algebraic expressions, to interpret such expressions, to make connections between concepts when solving linear equations, and to communicate algebraic concepts. Research indicates that students who used manipulatives in their mathematics classes have higher algebraic abilities than those who did not use manipulatives (Chappell and Strutchens, 2001).

John van de Walle and his colleagues (2013) define a mathematical tool as, “any object, picture, or drawing that represents a concept or onto which the relationship for that concept can be imposed. Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons).” While almost any object can be used as a math manipulative according to this definition, there are certain ones that seem to be more effective than others. Click on your grade level to see a list of recommended manipulatives.

4K-2nd Grades 3rd-5th Grades 6th-12th Grades

In today's environment, the use of virtual manipulatives has become more widespread. According to Moyer, Niezgoda, and Stanley (2005), "Virtual manipulatives are uniquely suited for teaching mathematics with young children. A web connection makes them free of charge and easily available. Some virtual manipulatives have the potential for alteration. For example, users can color parts of objects; they can mark the sides of a polygon or highlight the faces on a Platonic solid. This interactivity allows all children to be engaged in the problem-solving process. Some virtual manipulatives link symbolic and iconic notations by saving numerical information or providing mathematics notations that label the on-screen objects. The click of the mouse on many virtual manipulatives gives children access to unlimited materials. And clean up is easy - children simply click an icon and the on-screen objects disappear." Marzano has also done a lot of research looking at the parts of a child’s brain that light up for different activities that are done through technology. He has concluded the conceptual part of the brain can be tapped into by interacting with a visual image in a technological way.

A powerful strategy for using virtual manipulatives is to ask students to try and use as many different manipulatives as possible to solve one problem. For example, a third-grade teacher might ask students how they could use the base-10 blocks, color tiles, a hundreds board, and number lines to show 6 groups of 5. Have students work in breakout rooms to prepare their visual for showing their thinking. How do the different solution strategies connect? How are they different?

Click on the links below for some virtual manipulatives options.

Math Learning Center The Techie Teacher Dreambox

Toy Theater Didax Mathigon

NEW: SIS4Teachers List of VM (this allows you to sort by math concept)

Teachers cannot assume that when students use manipulatives they will automatically draw the correct conclusions from them. Instead, teachers need to bridge the manipulatives to the representational and then abstract understanding in mathematics so that students internalize their understanding. The best way to do this is to combine the concrete, representational, and abstract into the same lesson so that students can begin making the connections. In addition, highlighting all three of these phases at the same time allows your students to be able to work where they are within these phases.

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Marilyn Burns, a renowned math educator and author, believes we shouldn't teach without manipulatives. She shares her "7 Musts for Using Manipulatives":

1. I talk with students about why manipulatives help them learn math. These discussions are essential for first-time users and useful refreshers to refocus from time to time. I precede discussions by giving children time to explore a manipulative. Then we talk about what students noticed and I introduce the concepts they'll learn with the material.

2. From day one, I set ground rules for using materials. We talk about the similarities and differences between using manipulatives in class and playing with toys or games. With toys or games, children can make up their own rules. With manipulatives, they are given specific problems and activities. I do make clear, however, that they're free to make discoveries and explore new ideas.

3. It's also important for students not to interfere with one another. I step in when I hear a howl of protest as a student who needs one more yellow tile takes it from another group's table. Sometimes I open up the discussion to the entire class. These impromptu reminders help keep students on track.

4. I set up a system for storing materials and familiarize students with it. It's important for students to know where and how to store materials. A clear system makes the materials more accessible. Some teachers designate and label space on bookshelves. Others use zip-top plastic bags and portion materials into quantities useful for pairs or groups. Still others place a supply of each material at students' tables so they're always within reach.

5. Time for free exploration is worth the investment. Whenever I introduce a new material, I allot at least one math period for this. Teacher demonstrations alone are like eating a papaya in front of the class and expecting children to know how it tastes. Free exploration time also allows students to satisfy their curiosity so they don't become distracted from the assigned tasks. Expect children to see if tiles can fall like dominoes; build tall towers with rods; or construct rockets out of cubes. After children have explored a material, I ask what they've discovered and record their observations on a chart so their classmates can get insights from their ideas. Then I assign a specific task.

6. For easy reference, I post class charts about manipulative materials. Charts not only send students the message that I value manipulatives, but also help students learn materials' names and how to spell them. In September I post a chart that lists all the materials we'll use during the year. For some materials, I post separate charts to list their shapes and colors. And I leave posted charts of students' discoveries about materials.

7. Manipulatives are a natural for writing assignments. They provide concrete objects for children to describe.