Category: Teaching Math

The Taptana / Instructions Part 1

I was so fortunate to study with Rebeca and Mauricio Wild, two of the world’s foremost educators for child-based learning residing in Ecuador. During each of their seminars we worked with math hands-on learning materials, and one of the many materials they showed us was the Taptana, a calculator that is thousands of years old and was developed by the Inca people.
It consists of a wooden board with holes (there must be at least 10 holes) and beads in the Montessori colors. Here it is on the photo:


As you see on the photo below, each hole and each bead have a specific value:
Each green bead is worth  1
Each blue bead is worth 10
Each red bead is worth 100
Each yellow bead is worth 1,000
Each light blue bead is worth 10,000
Each orange bead is worth 100,000


Depending on where we place a bead its value changes.
For instance a green bead (worth 1) is worth 1 in the hole labelled 1,
it’s worth 2 in the hole labelled 2,
it’s worth 3 in the hole labelled 3,
it’s worth 4 in the hole labelled 4, and so on.

A blue bead (worth 10) is worth 10 in the hole labelled 10,
it’s worth 20 in the hole labelled 2,
it’s worth 30 in the hole labelled 3,
it’s worth 40 in the hole labelled 4, and so on.

A red bead (worth 100) is worth 100 in the hole labelled 1,
it’s worth 200 in the hole labelled 2,
it’s worth 300 in the hole labelled 3
it’s worth 400 in the hole labelled 4, and so on.

This way it works for each bead…for instance, an orange bead (worth 100,000) is worth 900,000 in the whole labelled 9. I guess you can see where this is going. Can you imagine the possibilities ?

Practice recognizing the value of beads simply by laying various beads on the board. On the photo below which value do you see?

Here’s the value for the left side: 526, 840

What’s the value on the right side? (solution below)

Solution: 9741

Instructions to the Montessori Square Root Board

The square root board shows in a simple way the essence of squaring a number and determining its root.

For both operations, children should already have multiplication experience. Taking square roots becomes very easy with the board, because one can see that is it the opposite of squaring. In order to square a number, it must be multiplied with itself: 2×2=4
One can see that this operation always looks like a square on the board; thus it is called “squaring”.

To find out the root of 4, we count the beads on one bottom –the root. There are 2 beads, which means the root of 4 is 2:

This works easily with all numbers from 1 to 9. Instead of beads, we can also use pebbles, blocks or buttons:

With numbers greater than 9, we need beads in various colors. Maria Montessori developed the following simple system:
Green = 1
Blue = 10
Red =100
Light green = 1000
Light blue = 10,000
Light red = 100,000

We see the board just like an x-y graph. Each bead on the x-horizontal line is multiplied with each bead on the y-vertical line. The result is placed in the right color at the point where the two lines meet. Here you can see what 12×12 looks like:

And these photos show the process step by step: 12 = 2 green beads for the ones and 1 blue bead for the ten (always place the lowest value beads in the beginning corner):

Then we place the same beads on the vertical line:

Then we can start multiplying: 1×1 =1 and 10×1=10 and 1×10= 10 :

10×10=100 and that’s the complete square:

To find out the result, we add the value of all beads: 4×1 plus 4×10 plus 1×100 equals 144. Here:

Even with large numbers, this technique remains simple. One can observe a specific variety of color patterns which increase with regularity. For example, here is 122 squared:

After all beads are placed, we can start adding: (1 x 10,000) + (4 x 1000) + (8 x 100) + (8 x 10) + (4 x 1) = 14,884

The root of 14,884 is the bottom line of the square, 122:

After we have learned to use the board to SEE the way squares grow, we can now learn the opposite process – finding the root of a number. First, we lay the square pattern using the appropriate amount of beads and then count one side.

The square root board helps to understand what the concept of square root actually means. It takes away the fear from this process usually only taught on paper.

I remember that I couldn’t understand square roots when I went to school and I am happy to have found the Montessori square root board! Thank you, Maria Montessori!
Try it!

Multiplication with children

Learning basic math becomes easy when we understand why it might be useful in real life!

Multiplication is really an expanded form of addition. We need it when we want to add several things of the same value. This can be communicated to children in informal ways:

It is needed in daily life, for example when cooking, e.g. we need 3 times 2 eggs to make 3 omelettes. We can count 1,2,3,4,5,6 or add 2+2+2, or multiply 3 times 2.

Or, when shopping e.g. 4 pieces of chocolate each cost $2. Instead of adding $2+$2+$2+$2 we can multiply 4 times $2.

When multiplication is associated with real-life situations, it becomes part of the child’s life experience, then the various kinds of doing multiplication on paper become useful and interesting.

Maria Montessori’s colored beads, the “multiplication bead bars” are the most useful learning / teaching material for the times tables.

The bars from 1 to 10 each have specific colors, which remain associated with the value of the bar.

At a glance, we can tell 5 times 3 is different from 3 times 5, even though the product is equal.

For example, we can lay out the 4 times table, and add the equation and product on small paper squares. This is not the best method to learn the times tables by heart, however, it is a great way of experiencing and comprehending the times tables, and learning how to use them in real life situations.

The Table of Pythagoras or decanomial boards: This material is made in the same colors as the beads, and at a glance we can see the geometric forms created by the times tables:
The multiplication of two equal numbers results in a square.
The multiplication of two different numbers results in a rectangle.

The complete Table of Pythagoras / decanomial board looks like this:

You can cut the complete Table of Pythagoras or a specific times table out of cardboard. decanomial cut-out

Enjoy and get creative!
Carmen Gamper

ENJOYING MATH: How Can Children Learn Math with Joy and Ease?

Translation of the April guest-post at Sybille Tezzele-Kramer’s Blog buntglas, originally in German and Italian “Wie findet mein Kind den Zugang und die Freude zur Mathematik?

In traditional schools math lessons can be confusing and scary for many children, and exhausting for teachers. Many grown-ups and elderly people remember their math lessons from childhood with aversion. Too many children and adults have been traumatized by math lessons. This fear results in confusion, low self-esteem, and aversion against the beautiful harmonious world of numbers. The joy and curiosity needed to explore the world of numbers gets lost when math remains an abstract topic on paper. It doesn’t have to be that way!

Basic math is not something that only exists on paper. On the contrary, it is a highly intelligent language, which developed out of the necessity to exactly describe the tangible environment and to orient ourselves in the world around us.

EVERYONE who is genuinely interested and curious can playfully learn basic math, because we can see it, touch it and live it. Often, on paper, math seems very complicated, whereas, when shown with real things, the same equation can actually be very easy to understand. In fact, the word mathematics derives from the Greek mathematikos, meaning eagerly learning, and scientific. Arithmetika means the art of calculating.


1. Mathematics is a universal language that connects ALL HUMAN BEINGS who ever lived on earth, because we all have similar experiences with matter. Thus, math serves as a tool of communication about specific phenomena (quantity, structure, space, and change) in our tangible environment – our body and home, our gardens and buildings, Nature…

2. Math empowers us to recognize, structure, change, and utilize things and patterns in our environment. It is a tool to consciously and intentionally shape and beautify items, and to build with them, live with them, and utilize them for our needs. Math is a language that empowers us to become creative and active.

ALL BASIC MATH CAN BE TOUCHED AND SEEN. All concepts of which we can hardly find examples in the world of matter are part of higher math.

Through the concepts and language (numbers and symbols) of basic math, we can exactly, quickly and easily describe the following phenomena in our environments:


• Countable Things (apples) are named with digits: 1,2,3, 1/2, 0.5

• Uncountable Things (flour) are named with measurements: 14kg, 82cm, 23%

• The Quantities of Whole Things (walnuts) are named with digits from 1 to infinite: 1, 2, 3, 1,000,000….

• The Quantity of Parts of a Whole (pieces of a cake) are named with fractions, percentages and  decimals: 1/4, 25%, 0.25

• Things We Don’t Have are named with negative numbers: -1, -5, -40

• Wholes and Parts of Wholes, when we add or take away, are described with addition and subtraction: 5+3; 8-2

• Wholes and Parts of Wholes, when we need more than once (each cake needs eleven strawberries) or we need to divide (each child wants the same amount of strawberries or pizza slices) are described with multiplication and division: 5×6; 30/10

• When a Specific Quantity is Unknown, or when the relationship between quantities is more important than the specific amount, then we can calculate with letters instead of numbers, which is called algebra: 30+b=35; a+b=c, c–b=a; a2+b2=c2


Parents and teachers can consciously develop awareness for math in their daily lives at home and at school. Try to find real-life occasions for observing, estimating, counting and measuring, or intentionally prepare such learning opportunities. Mathematical thinking starts with awareness for estimating, counting and recognizing patterns.

At home, in the kitchen and the garden, at the grocery store and on nature walks you can – together with the children – invent counting games. Those games, puzzles, and joyful competitions can be precious math lessons, without anybody even mentioning the word math, or writing numbers.



At schools, a kitchen can be very useful for math teachers! All basic math concepts are tangible in a kitchen and most children love to prepare their own food. Some innovative schools provide a kitchen for children to learn and experiment, such as the Rebeca Wild based (RWB)schools.


At home and at school, educators can prepare an intentional place to play grocery store, bank, and theater. In these environments we can introduce children playfully to math concepts, and then let them learn and explore in their own ways. Some innovative schools prepare role play landscapes for their students.

We can also provide hands-on learning materials, board games and a variety of kinesthetic math games developed by teachers from all over the world. Hands-on learning materials have been used for a long time on this planet. Some materials are thousands of years old, such as the Chinese abacus and the Incan taptana.

Maria Montessori developed the best math hands-on learning materials almost a hundred years ago.


The difference between paper math and real-life math is the deep understanding and the joy of learning that results from the fun of being actively involved in a mathematical activity. This involvement supports children in developing math as a tool they use in their personal life, and which makes their life easier (instead of harder). The insights gained from researching, experimenting, tinkering, and playing allow children to establish a personal relationship with their environment, and to feel self-assured in being active and creative. These experiences also build the ground for logical and creative thinking and solution-finding.


The intelligence which is developed through direct involvement remains connected to the body, and the abstract world of the mind is resting on the tangible world. These processes that have been presented support the intelligence of the heart. When the ability to think and reason is strongly connected to one’s own body, a human being is supported in the development of compassion, empathy and understanding of life and learning processes.

Let’s learn how to use math materials, and also inform educators about hands-on learning materials and ‘real life math’ in order to make children’s, parent’s and teachers’ lives easier.

Carmen Gamper can be invited to give basic hands-on introductions to math materials.


Mill Valley, California at the Harmony Montessori on April 29, May 25 and June 17 (each 7-9pm) and a weekend workshop in Honolulu, Hawaii, May 14 to 16, 2010.

Please learn more and register here:

In my next blogs, I will introduce my favorite math learning materials for you to use and create step by step.

Carmen Gamper

Math can be fun!…with hands-on learning materials!

Get inspired!! Math teacher, tutors and parents can make lessons way more interesting and accessible with hands-on learning materials. Children love to play and learn with the materials also after the lesson, in self-directed ways.

To see, touch and play with the ‘Times Tables’: The Decanom, developed by Dr.Maria Montessori

Learning decimals with REAL money!!

Deepening understanding for addition with Dr.Montessori’s beautiful Snake Game.

How much is 1000? How long is the pool? Experiencing length and width, and measuring real objects is exiting!

It is my joy to offer math tutoring for 6 to 14 year- old students with hands-on learning materials in Marin/San Francisco. Please write to References available upon request.

I utilize hands-on materials for addition, subtraction, division, multiplication for whole numbers, decimals and fractions, measuring materials, logic, strategy, pattern-recognition and fine motor development materials in order to give the abstract worlds of math a foundation in the concrete world around us.

Dear Math Tutors and Teachers: I offer consulting to introduce you to materials to help make your lessons experiential, efficient and fun. Contact me for a free introduction at

Best wishes,
Carmen Gamper