Category: Teaching Math (page 2 of 2)

Lattice Multiplication

Here’s a great way to teach multiplication! I find this is easier for beginning learners, than the traditional way.

The Montessori checker board is based on the same strategy:

Let’s make multiplication as a easy as possible for beginning learners! The traditional way can be shown when the essence of multiplying with large amounts is understood.

Best wishes,

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

•  Length, Width and Perimeters of Areas (a table) and solids (a room): square: 40cm x 4 = 160cm;  1 side of a square √160cm = 40cm; cube: 40cm x 40cm x 40cm = 64.000cm2 …

• 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 alternative schools provide a kitchen for children to learn and experiment, like the schools I co-created.


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

Many toy producers distribute Montessori materials without mentioning Montessori as the creator; it seems the Montessori materials have become common knowledge, and are now called “manipulatives.”


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.

With joy and courage,
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

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