This department recognizes the importance of children's exploring hands-on and minds-on mathematics and presents teachers with open-ended explorations to enhance mathematics instruction. These tasks invoke problem solving and reasoning, require communication skills, and connect various mathematical concepts and principles.

A mathematical investigation--

* has multidimensional content;

* is open-ended, with several acceptable solutions;

* is an exploration requiring a full period or longer to complete;

* is centered on a theme or event; and

* is often embedded in a focus question.

In addition, a mathematical investigation involves processes that include--

* researching outside sources;

* collecting data;

* collaborating with peers; and

* using multiple strategies to reach conclusions.

Investigations are somewhat structured in their sequence of activities. They come alive, however, through students' problem-solving decisions and strategies. Although having students follow a scripted sequence and set of directions for an investigation is possible, the NCTM Standards encourage teachers and students to explore multiple approaches and representations in their mathematical activities. As a result of their exploration, students also will use their reasoning and proof skills as they evaluate their strategies. The use of multiple approaches helps students find new ways of looking at things and understand different ways of thinking about a problem.

Introduction

An elementary student's query about how a six-dot notation, such as braille, could be used to represent the alphabet and numbers prompted the writing and field-testing of these activities. In this investigation, students explore patterns associated with coding the letters of the alphabet and numerals. "Preparing for the Investigation" features the reading and discussion of a book about Helen Keller and can be used at any grade level. Younger student investigators will explore coding systems with color tiles and investigate the use of braille to represent numerals. Older student investigators will extend the investigation of patterns and explore the braille alphabet.

Preparing for the Investigation

To prepare for the investigations, the student investigators should recognize the need for an alphabet that relies on the sense of touch, rather than sight, for people with visual impairments such as blindness. Reading a book about the life of Helen Keller--such as Helen Keller by Margaret Davidson (1997) or Helen Keller: Toward the Light by Stewart and Polly Anne Graff (1990)--can serve as a foundation for the activities. The "real" use of braille includes many shortcuts and conventions; however, these investigations use a simplified version of the numerals and letters.

Setting the Stage

Louis Braille was born in 1809 near Paris, France, and became blind when he was a young child. When he was fifteen years old, he invented a code for the blind to use to read. He found raised letters difficult to read, and a meeting with Charles Barbier inspired him to create a code. Barbier, a former soldier, showed Louis how soldiers had used a complicated twelve-dot system to communicate top-secret information on the battlefield.

Louis was able to design a simpler six-dot system that became the braille alphabet. The system features "cells" with three rows and two columns of dots that represent letters, numbers, and symbols. In this investigation, students will explore the patterns in braille's representations of numerals and letters. Students also will explore different coding systems, using colored tiles and dots to represent concepts, letters, and numerals.

Objectives

The student investigators will--

* identify and count combinations of colors to generate codes;

* explore patterns in color combinations;

* recognize braille as a code that employs a system of dots; and

* translate numerals to braille.

Materials

Each pair of student investigators will need--

* one "Colors, Codes, and Numbers" reproducible sheet;

* eight colored tiles: four of one color and four of another color;

* two markers that correspond to the two tile colors; and

Structuring the Investigation

1. After reading the book, explain to the student investigators that they are going to conduct an investigation to explore how they can create a code to represent numbers by using dots on a page.

2. Distribute eight colored tiles to each pair of investigators: four of one color and four of another; for example, four red tiles and four yellow tiles. Distribute a copy of the "Colors, Codes, and Numbers" sheet to each pair of investigators.

3. Ask the student investigators to arrange any three of their tiles in a row, or "train," and to record that train by coloring one of the trains on the reproducible sheet to match their tiles. Challenge the investigators to think of another choice of tiles or order of tiles for the train that is different from the first one. Have students color this result on the reproducible page. The student investigators should continue this process until they believe they have created every possible train for three tiles. Note that two of their solutions should include trains in which all the tiles are the same color.

4. The student investigators will use the "pairs share" cooperative learning strategy to find all the solutions. When the students in each pair believe that they have found all the solutions, they compare their findings with another pair's solutions. Investigators likely will ask, "Do we have them all?" Next, students use overhead tiles to share with the whole class the trains that they discovered. The students should be able to generate eight solutions; for example, RRR, YRR, RYR, RRY, YYR, YRY, RYY, YYY. Explain to the investigators that they could use each of these eight trains as a code or symbol to represent something or someone. For example, RRR could be a code for "dog," YYR could be a code for "school," and so on. Help students realize that the system is limited because it can represent only eight different things.

5. Ask the investigators to estimate the number of different symbols that they might create if they change the system to four tile slots instead of three. After discussing their responses, explain that their task is to use four tiles to form squares that have two rows and two columns, and to find all the symbols (patterns) that they can create. Using their colored tiles, markers, and number 2 on the reproducible sheet, pairs of investigators should try to generate all the solutions.

6. As in the "trains of three" exploration, after the pairs of investigators complete their work, they should compare solutions with another pair of investigators and then discuss the results as a class by using the overhead colored tiles. A recording system on the overhead or chalkboard can be used in which each solution is written as four letters (Rs and Ys) arranged in a 2 x 2 square. The investigators should discover that sixteen possible symbol combinations (patterns) exist when they use the 2 x 2 square. They also may be able to communicate that adding just one new "slot" to the code doubled the number of generated symbols.

7. Explain to the investigators how the braille coding system for the blind is based on a rectangle that has three rows and two columns--a cell with six "slots" for generating symbols. In the braille system, each of these slots may or may not contain a raised dot. A symbol that has a dot in all six slots is like a grid containing six tiles of the same color. A symbol with a dot in four slots is like a grid with four tiles of one color and two tiles of another color. This system makes it possible to create sixty-four different symbols. (Children in second and third grade do not need to show or prove this.)

8. Refer the investigators to the symbols for the ten digits on the reproducible sheet. Note that the "number" braille symbol is always placed before the digit symbols to notify the braille reader that a number is about to follow. Have the investigators complete the reproducible page by writing their ages in braille.

9. As an extension to the investigation, students can use braille to write solutions to word problems that they create or that the teacher creates, such as the following:

Each child in a class has 3 pencils. If there are 15 children in the class, how many pencils are there altogether?

Our answer written in braille: _________

Have several pairs of investigators share with the class the problems they created, and display their solutions as numerals and in braille on the board or overhead projector.

10. Conclude the investigation by explaining to investigators that the same code, which employs a 3 x 2 cell, is used to generate each letter of the alphabet so that the blind can read by tracing their fingertips over raised dots on a page. Remind them of Louis Braille, who developed this system more than one hundred fifty years ago. Ask the investigators to look for the use of braille in their lives. The symbols are common on everything from elevator buttons to optional braille menus in restaurants to plastic cup lids in fast-food restaurants.

Objectives

The student investigators will--

* explore combinations;

* identify and generalize patterns of numbers and symbols;

* use a table or exponents to determine numbers in a sequence;

* recognize braille as a code that uses a system of dots;

* translate words into braille; and

* invent a symbolic code.

Materials

Each pair of student investigators will need--

* one copy of the "Codes and the Alphabet" reproducible sheet;

* a calculator; and

* a necklace "string" and two boxes of beads of two different colors if they explore the extension activity.

Structuring the Investigation

1. Begin the investigation by reading a book about the life of Helen Keller, as suggested in the "Preparing for the Investigation" section. Remind the student investigators of Louis Braille's system for writing the letters of the alphabet and numbers and that the current system uses a series of dots placed in a 3 x 2 "cell." For example, a single dot placed in the upper left-hand corner of the cell represents the letter a.

2. Have the investigators discuss why they think that Braille chose to use a 3 x 2 grid size. Ask students why he did not keep the system simpler by using a 1 x 2 grid. Assuming that "no dots in the grid" is an option, the investigators should recognize that they can form only four characters by using a 1 x 2 grid; for example, both slots empty (EE), a dot only in the left (DE), a dot only in the right (ED), or dots in both locations (DD). Next, ask the investigators to determine how many different symbols they could create if Louis Braille had added one more "slot" in the code so that the code included three slots. During class discussion, the student investigators should be able to identify eight different symbols that are possible if they use three slots for the code (EEE, DEE, EDE, EED, DDE, DED, EDD, and DDD).

3. Have the investigators explore the number of patterns that are possible if they use a 2 x 2 square grid to create a code. Students might explore the problem as suggested in number 2 of the reproducible for grades 2-3 or discuss the problem with a partner and work as a class to generate all the possible codes (patterns). Student investigators should determine that sixteen different codes are possible when using a 2 x 2 square cell, assuming that the solution includes symbols with none of the slots filled and all the slots filled.

4. Distribute a copy of the reproducible sheet "Codes and the Alphabet" to each pair of student investigators. Ask the investigators to complete the table and question 1. Then, in a class discussion, the investigators should recognize a pattern: two slots produced four symbols, three slots produced eight symbols, and four slots produced sixteen symbols. They should have predicted that adding a fifth slot would generate thirty-two symbols. When the investigators conclude that each additional slot doubles the previous number of symbols, they should explain why this happens. The investigators should recognize that an extra slot doubles the answer by adding all the previous symbols with a blank slot and all the previous symbols with a dot-filled slot. The investigators then can predict that braille's six-slot cell can generate sixty-four symbols.

5. Have the investigators compare the number of letters in the alphabet with the number of slots needed for a braille-type alphabet. The investigators should acknowledge that the cell must have at least five slots because there are twenty-six letters in the alphabet. However, because the braille alphabet also must represent numbers, punctuation, capitalization, and so on, it needs more than twenty-six symbols. A 3 x 2 cell can generate sixty-four symbols, enough to capture the alphabet and numbers but still make reading with a fingertip manageable.

6. Have the investigators use a calculator to complete the next section on their reproducible page. As pairs answer these questions, have them discuss their responses. The investigators should discover that Barbier's system using a 6 x 2 arrangement of dots produced 4,096 different characters and was far too complicated to memorize and use for reading.

7. Have the investigators study the braille alphabet on the reproducible page. Ask them if they can find any patterns in the symbols. After they discuss the alphabet with their partners, have a class discussion. The investigators might recognize that the letters a through j establish a pattern; the letters k through t are exactly the same but with one dot added in the lower left corner of the cell. The rest of the letters are repeats of k through o but with one more dot added in the lower right corner of the cell. The only exception to this rule, and the reason that the letter w is separated from the others, is that w was rarely used in nineteenth-century France. Louis Braille saw no reason to create a symbol for the letter. The symbol for w was not added until after his death.

8. Using the braille letter code, investigators can encode their names in the space provided on the sheet. Then tell the investigators that a five-slot format technically would be sufficient to create a code for the alphabet. (The students already have shown, through patterns, that five slots would produce thirty-two symbols, and there are only twenty-six letters in the alphabet.) Have them create a new alphabet code that uses five slots in a circle, rather than six slots in a rectangular cell. Using the reproducible sheet, have each pair of investigators design a code for the twenty-six letters of the alphabet, making sure that they do not repeat any symbols.

9. Discuss with the students how they generated their symbols. Did they use a systematic or somewhat random process? Most of the investigators will discover that the simplest way to complete the task is to have a system, such as placing one dot in each of the five locations, then "fixing" one dot in a location and "moving" a second dot through the other four locations, then fixing two dots, and so on. This discussion is an excellent opportunity to remind the investigators that using a system--an organized way of listing data--will help them ensure that they have found all the possible solutions. The system underlying the braille alphabet is easy to learn because it involves repeating letters with one dot added, then repeating them again with another dot added. Physically recognizing patterns of raised dots and assigning them standard meanings, however, can be a challenging and time-consuming task, especially for students who also are learning to read.

10. Throughout this investigation, the student investigators are engaged in listing outcomes and counting possibilities. As an extension to the investigation, the investigators can make necklaces with beads. Each investigator should create a necklace in which two beads are one color and the remaining beads are another color. The students can place the two beads in any location relative to the other beads. Each investigator must create a unique necklace. The class will pursue the following problem: How many beads of the second color must the necklace include so that every investigator in the class has a different necklace? The student investigators can explore this mathematics problem and then construct and wear the necklaces.

11. Another extension that the class may want to pursue investigates the amount of braille that can be represented on a page. For example, braille pages usually are limited to twenty-five lines with thirty-six to thirty-nine cells on each line. Have the student investigators answer the following questions:

* About how many "characters" (letters, punctuation marks, and spaces) can fit on a page that is written in bra did you get your answer?

* Estimate how many characters are on a printed page of a book. How did you get your answer?

* About how many pages of braille would be needed to spell out one page of a printed book? How did you get your answer?

References

American Foundation for the Blind. "Braille Bug Site." www.afb.org/braillebug (3 Oct. 2002).

Ashcroft, S. C., and Freda Henderson. Programmed Instruction in Braille. Pittsburgh, Penn.: Stanwix House, 1963.

Davidson, Margaret. Helen Keller. New York: Scholastic Trade, 1997.

Doward, Barbara, and Natalie Barrage. Teaching Aids for the Blind and Visually Limited Children. New York: American Foundation for the Blind, 1968.

Graff, Stewart, and Polly Anne Graff. Helen Keller: Toward the Light. New York: Dell Publishing, 1990.

Lowenfeld, Berthold, Georgie Lee Abel, and Philip Hatlen. Blind Children Learn to Read. Springfield, Ill.: Charles C. Thomas Publisher, 1969.

Schubert, Leland. Handbook for Learning to Read Braille by Sight. Louisville, Ky.: American Printing House for the Blind, 1968.

Daniel Brahier, brahier@bgnet.bgsu.edu, teaches mathematics methods courses for early, middle, and secondary education majors at Bowling Green State University in Ohio. His research interests include the development of dispositions and the assessment of student progress in mathematics classes. In addition to his university responsibilities, he teaches an eighth-grade mathematics class at St. Rose School in Perrysburg, Ohio,

Edited by Alice Merz, amerz@isu.edu. Idaho State University, Pocatello, ID 83209-8059. This section is designed for teachers who wish to give students new insights into familiar topics in grades K-6. This material can be reproduced by classroom teachers for use with their own students without requesting permission from the National Council of Teachers of Mathematics Readers are encouraged to send manuscripts appropriate for this section to "Investigations," NCTM, 1906 Association Dr., Reston, VA 20191-1502.
COPYRIGHT 2003 National Council of Teachers of Mathematics, Inc.
No portion of this article can be reproduced without the express written permission from the copyright holder.