Category: Teaching Math (page 1 of 2)

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-directed 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 wholes (there must be at least 10 wholes) and beads in the Montessori colors. Here it is:


As you see on the photo below, each whole 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 whole labelled 1,
it’s worth 2 in the whole labelled 2,
it’s worth 3 in the whole labelled 3,
it’s worth 4 in the whole labelled 4, and so on.

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

A red bead (worth 100) is worth 100 in the whole labelled 1,
it’s worth 200 in the whole labelled 2,
it’s worth 300 in the whole labelled 3
it’s worth 400 in the whole 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? Comment below 🙂

Stay posted … soon I will publish a blog on how to do addition with the Taptana….or can you guess?

With smiles,
Carmen Gamper




Instructions to the Square Root Board

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

Then we place the same beads on the vertical line:

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

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 one side 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 could not understand square roots when I went to school and I am happy to have found the square root board!
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

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