Re-inventing Mathematics Education using Visible & Tangible Mathematics

Copyright Š 2010 Educational Solutions Worldwide Inc. First Edition Compiled by Educational Solutions Worldwide Inc. with contributions

by Amy Logan and Christopherr Mendoza.

Based on the works of Dr. Caleb Gattegno.

Algebricks™ is a registered trademark of Educational Solutions Worldwide Inc. All rights reserved Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

Table of Contents Overview .....................................................................................1 How it Works ...............................................................................3 Mathematical Situations ................................................................5 Curriculum ....................................................................................7 Background .................................................................................11

Overview

Visible & Tangible Math is an approach to teaching mathematics that focuses on the process for generating mathematics in the mind. While many people believe that math is the skill of a few gifted people, we believe that every student is capable of functioning like a mathematician. By taking an abstract idea like a number and representing it in a way that students can see and touch, we find that students can thoroughly understand mathematical concepts well beyond their designated grade levels.

1

Students begin by manipulating the Algebricks colored rods in particular ways, then describing the relationships between the rods in their own language. Students quickly come up with mathematical statements such as, “Two white rods are as long as one red rod.” Once aware of these relationships, students are introduced to mathematical language. The above statement can then become, “white + white = red,” or “2w = r,” or “1 + 1 = 2.”

The student is not memorizing these facts, but experiencing the reality behind the statement.

Take a look at the rods on the previous page. What relationships do you notice, and what equations can you think of to describe them?

Visible & Tangible Math gives students a mathematical experience that they can:

1

See clearly

2

Do easily

3

Never forget

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How it Works

As a general rule, Visible & Tangible Math uses mathematical situations to present challenges to students.

This means the rods and the questions are presented in engaging, game-like ways. By independently conquering these challenges, students expand their potential, increase their understanding of their own strengths, and set the stage for taking on bigger, more difficult challenges.

If math is presented to students as a body of facts to memorize, it will easily be forgotten. By experiencing mathematical challenges, and witnessing mathematical relationships, students truly believe and understand their conclusions, and can easily verify their answers. Rather than rely on the teacher to tell them if they are correct or incorrect, students can independently check the truth of a statement using the rods, or using knowledge gained from experience with the rods.

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How Math is made Easy Working through challenges generates first hand experience. With this experience, students can easily verify their answers – they do not have to rely on the teacher to know if they are correct.

Students first use their own words to describe the insights developed out of their experiences. When the time comes to switch to the traditional vocabulary of mathematicians, it is easily understandable.

The tools used to create situations include Algebricks, Geoboards, and The Number Array. These materials enable the creation of a rich and diverse range of mathematical situations that build upon each other.

Students can move from one area of study to another, while using the experience from the first to understand the next. For example, division can be derived directly from subtraction and does not require the knowledge of multiplication as is commonly thought.

4

Mathematical Situations Equivalence This situation helps students understand that 13 is not just one number, but can be a combination of many numbers through several processes. To create this situation, students might be challenged to make the length of 13 in as many ways as they can.

The Length of 13 6 twos + 1 4 threes + 1 3 fours + 1 2 fives + 3 2 sixes + 1 7+6 8+5 9+4 10 + 3 11 + 2 12 +1 5

Fraction of a Fraction Rather than start the process of multiplying fractions by memorizing rules and writing numbers, students begin by tackling a fun, game-like challenge. Their skill with finding equivalencies, and their ability to visualize colors and lengths, makes this a manageable challenge for young students.

3/5 x 5/8

3/8

1/2 x 3/5

2/3 x 6/7

3/10

4/7 6

1/6 x 3/4

1/8

Pythagoras’ Theorem Using the Algebricks, students can easily perceive the relationships in a right-angle triangle. By physically making squares, they can see the right triangle, and check if the two smaller squares truly equal the larger one.

c2 a2

b2

a2 + b2

=

7

c2

Square of a Sum In this example, the usual variables of a, b, c have been replaced with the first letter of the rod’s color name. This is how students would first come into contact with the concept of squaring a sum.

g+y b

= (g+y)2

b2

=

y

g g

g

(g+y)2

2

=

gy

y

g (g+y)2 = g2 + 2gy + y2 8

gy

g

y2

y

y

See it clearly, Do it easily, Never forget it.

Algebricks are a model of the rational number system. They can be used to quickly create a wide spectrum of situations which are: 1) mathematical, 2) visible and 3) tangible. When combined with an approach to teaching that emphasises the presentation of well crafted pedagogical challenges, mathematics becomes a subject students can see clearly, do easily and never forget.

Below are a sampling of the topics covered at a primary school level.

Visible & Tangible Math •

Sessions of free play

•

Number measure - study of numbers up to 10

•

Numbers up to 1,000

•

Procedures, algorithms for the four operations

•

Fractions as operators

•

Study of fractions

•

Different bases of numeration

•

Highest Common Factor and Lowest Common Factor

•

Squares, cubes, and cube roots

•

The set of integers

•

Length, area, and volume

•

Capacity and volume of round bodies

•

Weight, mass, and density

•

Simultaneous equations about directed numbers

•

Permutations and combinations

•

Sets and subsets of algebra sets

•

Arithmetic progressions and geometric progressions

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Working with the Learning Process Since Visible & Tangible Math focuses on the learner, we refer to the “learning process” rather than the “teaching process.” Below is a brief explanation of the steps the teacher will guide the student through in order to master a mathematical challenge.

1

2

Teacher helps students create a Visible & Tangible situation

Students are presented with a challenge

3

4

Working through the challenge, students explore, experiment and discover

Students describe the situation in their own language

5

6

Teacher introduces the language of math

Students practice to improve their skill

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Dr. Caleb Gattegno

Background Algebricks, also known as Cuisenaire Rods, were invented in the early 1930s by a Belgian primary-school teacher named Georges Cuisenaire. The rods were not widely used until Cuisenaire met Caleb Gattegno, a mathematics professor from the University of London, in 1953. Gattegno instantly recognized the rods’ value, and immediately began dedicating himself to developing the uses and applications of the rods. He provided a new teaching approach and a completely revised curriculum for mathematics. He realized that children are capable of much more than was traditionally expected, and developed his mathematics approach accordingly.

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Gattegno travelled the world demonstrating the rods, his own invention the Geoboard, and his approach to teaching. Providing a visible and tangible opportunity to learn about mathematics strengthens the students’ visualization skills, and allows them to easily do mathematics in their minds. This is why the Gattegno method of teaching mathematics is called Visible & Tangible Math.

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