Manipulatives

Manipulatives are physical objects that students touch, move and arrange to understand a concept, most often in math. Common math manipulatives include base-ten blocks, counters, fraction tiles, Cuisenaire rods, pattern blocks and algebra tiles. By working with real objects, students can see and feel what numbers and operations mean before they move on to written symbols. Manipulatives are used in general education classrooms, but they are especially important for students with math learning disabilities such as dyscalculia.

What are manipulatives?

A manipulative is any concrete object that represents an idea. Ten small cubes snapped together can stand for "ten." A fraction tile marked 1/4 shows exactly how big one fourth is compared to one whole. When students use manipulatives, they build understanding through hands-on learning instead of only memorizing rules and procedures.

Manipulatives can be store-bought math materials, or everyday items like buttons, beans, coins, straws or paper strips. Many classroom kits, Montessori math materials and homeschool sets are built around these tools. There are also virtual manipulatives: interactive, digital versions of blocks, tiles and number lines that students move on a screen, for example on a tablet, an interactive whiteboard or in Google Slides. Virtual manipulatives are useful for remote learning, for students with weak fine motor skills, and for showing a model to the whole class. Many teachers use both physical manipulatives and technology, depending on the lesson.

Examples of math manipulatives

• Base-ten blocks: Small cubes (ones), rods (tens), flats (hundreds) and large cubes (thousands). They teach place value, regrouping, addition, subtraction and decimals.
• Counters: Small chips, bears or two-color counters used for counting, sorting, one-to-one correspondence, simple addition and subtraction, and integers.
• Fraction tiles and fraction circles: Pieces that show halves, thirds, fourths and more. Students compare fractions, find equivalent fractions and add fractions.
• Cuisenaire rods: Colored rods of 10 different lengths, each representing a number from 1 to 10. They are used for number relationships, measurement, fractions and ratios.
• Algebra tiles: Tiles that stand for x, x-squared and 1. They help older students model expressions, combine like terms, solve equations and factor.
• Pattern blocks: Geometric shapes for patterns, symmetry, geometry and fractions.
• Decimal squares and money: Grids and coins that help students visualize tenths and hundredths.
• Linking cubes and counting cubes: Snap-together cubes for counting, patterns, measurement and multiplication arrays.
• Dice and spinners: Used in math games to practice addition and subtraction facts in a fun, low-pressure way.
• Ten frames and number lines: Tools that help students see numbers and their relationships.
• Geoboards, clocks and play money: Used for shapes, telling time and money skills.

The concrete-representational-abstract (CRA) approach

Manipulatives are often used as the first step in the concrete-representational-abstract sequence, also called CRA (or concrete-pictorial-abstract). This is a well-known teaching approach for math, and it is widely recommended for students who struggle in math:

1. Concrete: Students solve problems with real objects. For 3 x 4, they make 3 groups of 4 counters.
2. Representational: Students draw pictures, tallies or diagrams to show the same idea. They sketch 3 circles with 4 dots in each.
3. Abstract: Students use only numbers and symbols: 3 x 4 = 12.

The goal is to build a bridge from hands-on understanding to written math. Teachers move to the next stage when the student shows understanding, and they can go back to the concrete step whenever a new or harder concept comes up. The What Works Clearinghouse, part of the U.S. Department of Education, recommends in its practice guide on math intervention that teachers use a well-chosen set of concrete and semi-concrete representations with students who struggle in math.

Manipulatives for students with dyscalculia and learning disabilities

Students with dyscalculia or another specific learning disability in math often have trouble with number sense, place value, math facts and the meaning of operations. Manipulatives can help because they:

• Make abstract ideas concrete and visible.
• Reduce the load on working memory, since the objects hold the quantities.
• Let students check their own thinking by counting or comparing.
• Build real understanding instead of memorized steps that are easy to forget.
• Give students a way to show what they know without heavy writing.
• Support problem solving, for example acting out word problems with counters.
• Add a sensory, hands-on element that keeps many students engaged.

Special education teachers often use manipulatives as part of specially designed instruction in math. The use of manipulatives can also be listed as an accommodation in an IEP or 504 plan, for example "access to manipulatives for math classwork and tests." Whether they are allowed on state tests depends on the state's testing rules. Learn more in accommodations vs. modifications.

Manipulatives also support students with intellectual disabilities, ADHD and autism, many of whom learn well through hands-on work. Students with fine motor difficulties may need larger pieces, magnetic tiles or virtual manipulatives. An occupational therapist can help choose tools the child can handle.

Reading manipulatives

Manipulatives are not only for math. In reading instruction, students use hands-on tools to learn how sounds and letters work:

• Letter tiles and magnetic letters: Students build, change and read words, which supports decoding and spelling.
• Sound boxes (Elkonin boxes): Students push a token into a box for each sound they hear, building phonological awareness.
• Syllable cards and word-building cards: Used to break longer words into parts.

Multisensory reading programs used with students with dyslexia often include these kinds of tools.

Tips for using manipulatives well

• Connect the objects to the math idea out loud. The tool only helps if students understand what it stands for.
• Give students time to explore the materials before a lesson.
• Move from concrete to pictures to symbols, following the CRA sequence.
• Do not remove manipulatives too early. Some students need them longer than others.
• Use them at home, too. Coins, dried beans or LEGO bricks work well.

Frequently asked questions about manipulatives

What are manipulatives in math?

Math manipulatives are hands-on objects, such as base-ten blocks, counters, fraction tiles and algebra tiles, that students move and arrange to understand math concepts like place value, operations and fractions.

What are virtual manipulatives?

Virtual manipulatives are digital versions of math tools that students move on a computer or tablet. They work like physical tools and can be helpful for remote learning or for students with fine motor challenges.

Are LEGO bricks manipulatives?

Yes, they can be. Any object that students use to represent a math idea can work as a manipulative. LEGO bricks are often used at home to show place value, fractions or multiplication arrays.

Are manipulatives only for young children?

No. They are common from kindergarten through the elementary grades, but algebra tiles, fraction tiles and geometric models are also used in middle school and high school. Any student learning a new, abstract concept can benefit from a concrete model.

Why are manipulatives important in special education?

They make abstract ideas concrete, reduce memory demands, and give students with learning disabilities a clear path from understanding to written math through the CRA approach.