Hands-on math: understanding multiplication

September 2, 2018

The other day we were checking if Nat knew the times tables. I never asked her to memorize them, so I was positively surprised when it turned out she knew most of the facts. Those that she did not know, she was able to calculate fast. With one exception and xceptions are opportunities to learn. So we sat down, took the Spielgaben sticks and checked if her method was correct or not (it was not) and why.

Children are often asked to memorize times tables, not knowing what lies behind the numbers they are trying to remember. However, a good understanding of multiplication is crucial for more advanced math.

Children learn better when we introduce a new concept using manipulative materials. It helps them visualize the problem they are trying to solve. They will also have a better understanding when they worked on the same concept using different learning materials.

When a child has understood multiplication, she alone will move from using manipulatives to do the mental calculations. She will develop her strategies for finding the result. And the memorization will happen alone along the way.

hands-on math: multiplication

Below are some examples of how to enhance the learning process by using manipulatives.


Use any kinds of counters to show the grouping concept. We make groups of several objects and show how to write down the multiplication fact.

hands-on multiplication
hands-on multiplication

Repeated addition

We use multiplication when we would have to add the same amount of objects repeatedly. We change counters to keep a child engaged.

hands-on multiplication
hands-on multiplication

Since we use Montessori materials at home, we often do multiplication with bead bars. It is a very visual and elegant material (you can find a full presentation on Montessori Album)

hands-on multiplication
hands-on multiplication


An array is an arrangement of similar objects, pictures or numbers in rows and columns. At home, we often make arrays with Lego® bricks.

hands-on multiplication

Montessori Multiplication Board

One of my favorite math materials. It is based on the concept of arrays. If you do not own one, it is simple to prepare. Working with the board explains how a times table was created.

hands-on multiplication
hands-on multiplication

When we were traveling and had no access to our Montessori materials, we made a board with tiny Lego® pieces.

hands-on multiplication

Times tables - where do they come from?

We used Montessori Decanomial Bead Bars to show where do the 1-10 times tables come from.

hands-on multiplication
hands-on multiplication

Properties of multiplication

When introducing properties, it is also useful to visualize it with manipulatives. It will help the child understand the basis of the concept and create effective calculation methods.

Commutative property: a x b = b x a

hands-on multiplication

Associative property: (a x b) x c = a x (b x c)

To explain this property we used Blokus 3d - we calculated how many cubes made the prism. It is easy to notice how the order of factors does not change the result (3 x 5) x 4 = 3 x (5 x 4).

hands-on multiplication

It can also be shown with flowers, more precisely, with the total number of petals. Whether we have five bouquets of four flowers, each with three petals or three bouquets of five flowers, each with four petals, we get the same result.

hands-on multiplication
hands-on multiplication

Distributive property: a x (b + c) = a x b + a x c

  • with Montessori bead bars
hands-on multiplication
  • with Lego® bricks
hands-on multiplication

Which manipulatives to use?

  • Montessori and Montessori-inspired math materials
  • all sorts of counters: plastic chips, buttons, stones, beans, glass pebbles, plastic crystals, board game pawns, etc.
  • brand counters - at home we have different sets from Learning Resources
  • Lego® bricks (yes, one of our favorite manipulatives)
  • wooden resources/blocks - we use Spielgaben (Froebel gifts)
  • interlocking cubes
  • stickers
  • playdough
  • and more

From manipulatives to mental calculation

Younger children learn better when first presented with physical objects, then pictorial representations and only then with the numbers. However, the older the child, the more she has practiced, the faster she is moving from using manipulatives to mental calculations.

Nat has developed her own strategies to calculate multiplication problems. Some of them take less time, and others take more. Although I never asked her to memorize the tables, she can quickly recall most of the facts. The memorization process has happened alone.

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