2 Math Manipulatives That Will Solve Your Math Problems

"What the hand does, the mind remembers."Maria Montessori

Research shows that over 7% of students experience severe learning disabilities in math, and countless more struggle because of the way mathematical concepts are presented to them. Our goal at ShillerLearning is to make math understandable for every child through hands-on, exploratory learning.

🧮 What Montessori says about concrete manipulatives and number sense: Maria Montessori designed her decimal system materials specifically to make the abstract concrete. She believed that children cannot truly understand place value, operations, or number sense until they have physically experienced them with their hands. Her materials, including unit beads, ten bars, hundred squares, and thousand cubes, allow children to see, touch, and manipulate the very structure of our number system. Research on dyscalculia and math intervention (Butterworth, Varma, and Laurillard, 2011, Science) found that children with math learning difficulties show the greatest gains when instruction uses concrete, physical representations of number rather than abstract symbols alone. A 2017 study in ZDM Mathematics Education confirmed that structured use of base-ten manipulatives produced significantly stronger place value understanding and operation fluency than symbolic instruction, with benefits extending to students with and without learning differences.

Is your child struggling with math, or would you like to prevent math difficulties from developing in the first place? Decimal material (also known as base ten blocks) paired with Montessori number cards are a game-changer for students young and old. Together these hands-on tools make the decimal system and place value visually obvious and number sense easy to acquire. They provide a concrete introduction to the decimal system in an amazingly understandable way and can be added to any math program.

Manipulative 1: Decimal Material

The decimal material physically represents each place value category. There are unit cubes (1 cm x 1 cm x 1 cm), ten rods (1 cm x 10 cm x 1 cm), hundred flats (10 cm x 10 cm x 1 cm), and thousand cubes (10 cm cubed). Used in sufficient quantity, these math manipulatives not only help children visually quantify each place value amount, but also provide hands-on learning for inequalities, multi-digit operations (addition, subtraction, multiplication, and division), exchanges within operations (borrowing and carrying), skip counting, and more.

Montessori decimal material base ten blocks

The decimal material is introduced using the Montessori three-period lesson, a simple and effective approach that ensures the student understands the concepts and materials being used. The ShillerLearning YouTube channel features videos on the three-period lesson, as well as performing operations with the decimal material and number cards.

First, the units are used to introduce and strengthen recognition of quantities one through nine. The materials are then paired with number cards to learn the symbols, the numerals 1 through 9.

Manipulative 2: Number Cards

The number cards help children recognize the numerical symbols for each place value category of the decimal system. One of the significant benefits of the number cards for students with special needs is that they empower students to work with numbers without needing to write them. The number cards consist of four sets for the first four place value categories. The smallest cards are numerals 1 to 9 (units), increasing in size for place value with numerals 10 to 90 (tens), 100 to 900 (hundreds), and 1000 to 9000 (thousands). The cards are color-coded in green, blue, and red, with the color pattern repeating for each new place value family.

With success achieved in numbers one through nine, students are ready to learn place value into the thousands. As they work with the material, they also discover that counting beyond nine in any place value category means continuing into the next highest category. Abstract, sometimes difficult concepts become hands-on, concrete, and interactive. This introduction and exploration of numbers is unparalleled in creating a solid number sense and understanding of place value.

4-Digit Addition with Exchange: A Step-by-Step Walkthrough

The following is a summary of activities scripted in the ShillerLearning Math Kit I, Lesson Books 1 and 2. Students first learn 4-digit static addition (without an exchange), then learn the exchange game, and finally move to dynamic addition with carrying.

Step 1. The problem is constructed on the work mat with the number cards and addition sign.

Step 1: construct the problem with number cards

Step 2. The decimal material for the first addend is laid out with the number cards. Repeat for the second addend.

Step 2: lay out decimal material for each addend

Step 3. Bring down the units (ones) and lay the 9 number card below them.

Step 4. Bring down the tens. In this case, we have 13.

Step 5. Trade 10 of those tens for 1 hundred flat, keeping the remaining 3 tens.

Step 6. Show that the hundred flat goes above the hundreds column, just as it will appear abstractly with pencil and paper. That floating carried digit, which confuses so many students, is no longer mysterious. They have experienced exactly what it represents.

Step 7. Bring down the hundreds, lay out the 500 number card, bring down the thousands, and lay out the 3000 number card. Finally, read the problem as a whole by overlapping the cards: 2192 plus 1347 equals 3539. A ruler may be used as the equal sign bar.

Here is a video that pulls it all together with a different example.

You will be amazed at how quickly young children achieve conceptual understanding and success in performing operations. The activities are also a powerful remediation for older students struggling with math for a variety of reasons, and crucial support for students with special needs. The number cards and decimal material are available separately and can be added to any math program. Both are included in Math Kit I, which covers Pre-K through 3rd grade.