Logo is a unique piece of software. In the purest sense, Logo is a programming language; it is a full-featured computer language derived from LISP, the language of artificial intelligence. More important, however, Logo is a language for learning. It is the right tool to teach the process of learning and thinking. Logo provides an environment where students assume the role of teacher. As a teacher, they must:

• understand the knowledge to be taught
• plan an approach to impart this knowledge
• break the knowledge into small, understandable chunks
• know how to clearly communicate the knowledge
• establish this new knowledge as the foundation for future learning
• be aware of and build on knowledge that the learner already has
• be receptive to exploring new ideas as they appear
• respond to the learner's (computer's) misunderstandings and errors

Students do this in Logo by:

• experimenting with Logo commands to understand them and be confident in their use
• planning their task and organize it into its various components
• writing a set of instructions to perform each small task
• constructing a program to perform all the tasks in the right order
• evaluating their program to assess whether the task is performed correctly
• debugging their program by locating and correcting errors or restructuring their approach

Logo is not limited to any particular topic or subject area. However, it is most useful for exploring mathematics, as Logo's turtle graphics provides a natural mathematical environment. Since the turtle moves in distances and turns in degrees, studying geometry by constructing and investigating polygons and figures makes Logo a powerful learning tool.

Designed at MIT as a language for learning, Logo is by its nature:

friendly	Logo is easily grasped; we can relate to the turtle and use it as an object to think with.
extensible	Logo can be taught new commands and other commands can be built thereon.
forgiving	Logo offers immediate feedback through helpful and informative messages.
flexible	Logo is as useful with preschoolers as it is with students of higher mathematics.
powerful	Logo is a programming language, providing all the tools needed to create programs of any degree of sophistication.

So although Logo is a programming language, it is better thought of as a language for learning; a language that encourages students to explore, to learn, and to think.

Logo's best known feature is the turtle, a triangular cursor used to create graphics. Even young children quickly learn to move and turn the turtle using easily-remembered, intuitive commands. For example, typing forward 50 moves the turtle forward 50 pixels (screen dots). Typing right 90 turns the turtle right (clockwise) 90 degrees.

By combining these commands, it is easy to draw a square.

forward 50 (You can also abbreviate forward as fd.) right 90 (You can also abbreviate right as rt.) forward 50 right 90 forward 50 right 90 forward 50 right 90

Using repeat, you can combine commands to form patterns. Here is the same square design drawn using one instruction line:

repeat 4 [forward 50 right 90]

Because Logo is an extensible language, you can add new commands by creating short programs or sets of instructions called procedures. For example, here is the procedure that will draw our familiar square:

to square repeat 4 [forward 50 right 90] end

Now, to draw a square, simply type square. You can use the word square as you would any other Logo command, even including it in other procedures. Procedures are the building blocks of larger programs. For example, here are some ways you can use the square procedure.

You could draw a flag: forward 60 square back 60

You could make a circle of squares: repeat 12 [square right 30]

You could combine a square and a triangle to build a house (typing house to draw it).

to house square forward 50 right 90 triangle end to triangle repeat 3 [forward 50 left 120] end

You can also use a name to represent the size of the square. to sq :size repeat 4 [forward :size right 90] end Now you can draw squares of different sizes by typing sq 10, sq 20, sq 30, etc.

The design below was created using two procedures that are more complex. They use procedure inputs to represent the line lengths and turning angles, recursion to call the same procedure again, and a conditional statement to make the procedure stop. You would enter the name of the main procedure design to run the program.

to design clearscreen right 30 polyspi 5 120 end to polyspi :size :angle if :size > 205 [stop] forward :size right :angle polyspi :size+5 :angle+.12 end

As you can see, just using turtle graphics, you can progress from drawing simple shapes with easy-to-learn commands to creating complex figures using quite sophisticated programming techniques. While turtle graphics are an excellent way to begin to learn Logo, you should view them as an introduction and building block, not as the end of a learning adventure. There is so much more that you can do with Logo!

Logo is a classic. Although it was one of the first pieces of educational software available, it is not outdated.

Professor Seymour Papert, designer of the Logo language, tells us what is important about Logo in his book, Mindstorms: Children, Computers and Powerful Ideas (p. 6).

"Two fundamental ideas run through this book. The first is that it is possible to design computers so that learning to communicate with them can be a natural process, more like learning French by living in France than like trying to learn it through the unnatural process of American foreign-language instruction in classrooms. Second, learning to communicate with a computer may change the way other learning takes place....We are learning how to make computers with which children love to communicate. When this communication occurs, children learn mathematics as a living language. Moreover, mathematical communication and alphabetic communication are thereby both transformed from the alien and therefore difficult things they are for most children into natural and therefore easy ones. The idea of "talking mathematics" to a computer can be generalized to a view of learning mathematics in "Mathland" that is to say, in a context which is to learning mathematics what living in France is to learning French."

Since Logo's beginnings in the mid-1960s, versions have evolved that take advantage of newer hardware and software, but the Logo philosophy and the basics of the language remain the same. Unlike other pieces of educational software, Logo is not based on a period of time, a geographical place, or current styles and trends. A timeless piece of software, Logo is as worthwhile now as when it was introduced. After more than a decade of Logo use in schools, a 1990 survey of past and current teachers of Logo showed that an amazing 98% believed that Logo was still appropriate in the classroom.

But what's the point of Logo? Why is Logo so important to learning? Logo involves more than just moving a turtle or using mathematics. The essence of Logo involves thinking about processes; about how you are doing what you are doing. In Logo, the process of creating your product is more important than the finished product. It is more interesting and educational to look at how a design was created than to look at the design itself.

Although designed for children, Logo is now used from kindergarten to the university. Logo is easy to learn, yet offers power to the experienced user; it is said to have "no threshold and no ceiling." Thus, students who can barely read can learn with Logo, while experienced computer users find that Logo provides a powerful and flexible programming environment.

To give you an idea of who uses Logo,

• First graders in New Hampshire use a single-keystroke version of Logo to move the turtle and explore shapes and lines.
• Second and third graders in Arizona learn about polygons using Logo.
• Fourth graders in California program a miniature golf game in Logo.
• Fifth graders in Massachusetts learn the geography of their state by drawing a map in Logo.
• Middle school students in Kentucky use Logo to control robotics.
• High school girls conquer "mathphobia" in a summer program at Mount Holyoke College in Massachusetts using Logo.
• Secondary students in California use Logo to create scatter plots in their statistics class.
• Secondary students in Colorado learn to program using Logo.
• Paralyzed students in Pennsylvania use a single-switch device with Logo to move the turtle and create designs.
• A professor at the Massachusetts Institute of Technology uses a special version of Logo to teach music theory.
• An instructor at the University of California at Berkeley teaches computer science students how to program using Logo.
• Students all around the world use Logo to learn.
  

Logo fits easily into the mathematics class, but it can also be used to explore other areas. The focus in most classrooms is not on programming, but on the thinking processes that Logo encourages.

mathematics:

estimation	working with distances and angles
polygons	using REPEAT to create regular shapes
perimeter and area	investigating number relationships
symmetry	drawing with point and line symmetry
coordinates	plotting points and graphing lines
probability	using Logo's random number generator
functions	writing functions that output values
algebra	graphing linear and quadratic equations
geometry	drawing and measuring lines and angles
trigonometry	using Logo's sine and cosine functions
fractals	combining graphics and recursion

programming:

proper techniques	writing structured programs
program design	breaking down a problem into smaller tasks
flow of control	learning about branching and conditionals
variables and recursion	using the power of the language
data handling	manipulating numbers, words and lists

language arts:

sentence structure	generating Logo sentences that follow the rules of grammar and parts of speech
creative writing	writing and illustrating poems
word structure	writing programs that rhyme words, pluralize nouns, and conjugate verbs

social studies:

directions	translating the turtle's heading into compass points
cartography	making maps using Logo graphics
foreign languages	creating foreign language command names

science:

robotics	controlling robotics devices through Logo
sensors	attaching light and touch sensors to the computer and reading the output
simulations	running physics experiments

fine arts:

computer art	using Logo's graphics capabilities
music	using Logo's sound-generating abilities
dance	choreographing the turtle
multi-media	capturing Logo graphics on videotape

Teachers report that Logo offers students many measurable gains (understanding of angles, improved estimating skills, etc.). But Logo also helps students with personal development, attitudes toward learning, depth of understanding, creativity, and other long-term benefits. Teachers are enthusiastic about teaching with Logo, and students about learning with Logo.

Logo gives students:

Classroom teachers offer these comments about Logo:

"It is a blank slate upon which students build at their own pace and at their own skill level." "There is so much learning going on when kids use Logo. It's the kind of learning they can't get any other way. I see something magical happening when kids use Logo..." "It presents all students with the chance to succeed in a non-threatening environment. Kids see that there are many ways to solve a problem. They also learn to cooperate and honor others' skills." "Students can create and express themselves." "It can be an integral part of a good curriculum to teach kids how to think and discover."

Classroom teachers experience the "magic" of using Logo, and believe that it helps them teach their students in many different ways. Researchers ask if this intuitive belief that Logo is an excellent teaching tool can be quantified. That is, do children using Logo exhibit statistical gains over non-Logo using students?

Studies have shown that students who use Logo:

• plan more efficiently
• represent planning tasks differently
• have an increased understanding of geometry
• persevere in solving problems
• are better at resolving conflict
• are more self-directed
• exhibit desirable social interactions

Some studies have had mixed or inconclusive results. Why? Possibilities include:

• teachers' lack of familiarity with an appropriate teaching style
• unaddressed critical variables, such as instruction method used, and "cultural climate" in the classroom
• the fact that gains may not be measurable within the timeframe of the study
• the individual teacher
• the level of instructional support provided
• the fact that skills acquired using Logo may not transfer to particular mathematical insights or cognitive gains without instructional direction

Doug Clements summarizes Logo research results (Logo Exchange, January 1988):

Logo's potential to develop geometric ideas will be fulfilled to the extent that teachers help shape their students' Logo experiences. Students do not automatically transfer knowledge gained in one situation to another. Repetition is not sufficient. Questions that cause students to reflect on what they were doing are instrumental.

One conclusion is clear:

The key to learning is the teacher. Logo is a good tool, but not one that most students will use to advantage on their own. Good teachers involve students in learning, give them choices, create a helpful environment, and turn mistakes into opportunities for exploration. Logo is the perfect tool for these teachers, as its very design makes it friendly, extensible, forgiving, flexible and powerful. These attributes are built into the language, making Logo an intrinsically powerful tool for learning.

The teacher is most important to students' learning. There are many approaches to teaching Logo, or guiding or facilitating Logo learning.

Often, the student creates his or her own task, one that is personally motivating and challenging. The classroom teacher then serves as a facilitator, helping students understand new Logo commands that might be useful; suggesting approaches to the task; helping determine where and why things are not going as expected; and offering support, suggestions and encouragement, when needed. Logo is a participatory, hands-on environment where both student-teacher and student-student interactions are important.

Logo author and educator Donna Beardon explains what is needed:

"Effective Logo teachers walk a fine line between free exploration and preplanned curricula: Logo is powerful, but Logo is not magical. Insights are powerful, but insights are not magical. Insights occur as a result of trial and thinking. Insights are what Logo is so good at promoting if the ingredients are all there. These ingredients include: 1. Steps toward understanding; 2. Significant problems that challenge and entice students to think hard; 3. Time to get in tune with the activity."

A teacher of Logo must know how much information to give, so that the student is able to discover important concepts. When students discover concepts, they "own" that knowledge. They have a much deeper understanding than if the teacher told them or even showed them.

Dan and Molly Watt, noted Logo specialists and authors of Teaching with Logo, note that individuals rarely complete projects alone, with no input, advice, or information from anyone else. Logo projects should be organized as real-life projects, where there is constant communicationówith peers and with advisors. Students working together learn Logo, encounter new problems to solve, and learn to work cooperatively. They are encouraged to share their knowledge, insights, discoveries, and strategies.

The Watts encourage students and teachers who are beginning with Logo to keep a journal of their project as part of the learning experience. In their journals, students record how they chose their project, new commands that they learned during the project, important discoveries that they made, problems that they ran into and how they solved them, and how they tested and debugged their program.

The final stage of a project is to publish it or put it in a form to share with others. This completes the project and gives the students pride in their work. Publication might mean sharing projects within the school or with students in a pen-pal school, creating bulletin board displays and library exhibits, writing articles for school or local newspapers, making a parent's night presentation, or sending projects to computer education magazines and journals for publication.

Logo provides a wonderful opportunity for parents and children to work together. If your home computer is the same type as the school's computers, you should try to get the same version of Logo for home. Logo manufacturers usually have arrangements so that students can use Logo at home for a reasonable cost. If students can use the same version in both places, it will be easy to take a copy of a project home to work on.

If the computer at home is a different brand, there is most likely a similar version of Logo available. Most Logo languages use the same fundamental turtle graphics commands. Differences may occur in commands that involve color and disk functions, and the syntax of the commands may be different in some cases. Students are quite flexible, however, and usually have little difficulty using a version of Logo that differs slightly from the one with which they are familiar.

Parents can also get involved with Logo at their children's school. First, find out if your school has the Logo language and will welcome your volunteer help.

If your school already has Logo:

• find out if your child's class is using it
• get Logo to use at home, if you have a computer
• help the parent-teacher organization arrange for teacher training
• suggest a session for parents and children to introduce them to Logo
• volunteer in the classroom to give teachers extra time to plan for and use Logo
• help prepare displays of student Logo projects

If your school doesn't have Logo, or isn't using it:

• share this information with the teachers and administration
• suggest that they use Logo to teach problem solving and math
• arrange for a Logo-using teacher in the area to make a presentation
  

Logo is one of many pieces of software that schools use to teach. How does Logo compare with other types of software?

Also known as CAI (computer assisted instruction), drill and practice programs provide practice in specific skills, such as multiplication or spelling. These programs may be appropriate for some students at times, but unlike Logo, they typically:

• control the student
• present problems with a single correct answer
• get predictable, causing students to become easily bored
• provide inappropriate feedback to students
• allow for minimal customization by the teacher
• make unnecessary and distracting use of sound and graphics

Simulations and problem solving software set up an experiment or situation that the student can control and explore. Well-designed programs in this category offer a valuable experience similar to that offered by Logo. Unlike simulations that could be written in Logo, however, these programs may restrict the user's ability to make changes to the content, the conditions of the experiment, or the algorithms used to determine results.

Teachers find that tool programs such as word processing, database, and spreadsheet programs provide excellent opportunities for learning. In a sense, Logo is also a tool program, useful by both students and teachers. When students use a tool program, either a word processor or Logo, they are provided with an environment

• that they can control completely
• that allows them to begin, revise, refine and complete a project
• that is useful for students in any grade and at any level
• that has a minimal amount of knowledge necessary to begin
• offers appropriate and nonjudgmental feedback
• can be used in any subject area

If your goal is to teach the process of thinking, give your students a hands-on environment for mathematics explorations, or combine the capabilities of several types of tools, then Logo offers the power and flexibility you need.

Although Logo is most frequently used by children who only explore a small portion of the language, Logo is a full-featured programming language. Almost any program can be written as well in Logo as in any other language, and frequently more quickly and elegantly. Logo is the ideal first language to learn because:

Logo is an easy language to learn.

• the commands are English-like and easy to remember
• there are friendly and helpful error messages
• it is interactive, so the user gets immediate feedback

Logo promotes good programming habits.

• it encourages the development of short programs (procedures)
• procedures are then used as building blocks by other procedures
• there are no line numbers that encourage jumping around the code

Logo allows you to create and reuse procedures.

• as an extensible language, Logo allows you to add new commands
• you can run and test procedures independently
• it is easy to use procedures from one program in another
• it encourages the creation of a library of often-used procedures

Logo programs are easy to write and maintain.

• programs are easy to debug with tracing and pausing tools
• you can use long and descriptive variable and procedure names
• error messages are detailed and helpful

Experienced programmers should be aware that:

• Logo is derived from LISP, the language of artificial intelligence
• Logo is a recursive language
• Logo offers dynamic variable scoping
• Logo offers a full range of disk and file handling commands

Logo is structured similarly to Pascal. Program development is done in the same way: you create a main program, which calls other procedures and functions, passing them variables as needed. Both languages are recursive.

If you would like to learn more about Logo, find ways of using it in the classroom, improve your programming skills, and meet other Logo-using educators, here are some ideas for resources you might find in your area. Details about specific resources to contact are included in the directory that follows.

• Attend in-service training provided by your school district.
• Investigate courses or summer seminars at colleges in your area.
• Inquire about Logo training from your Logo manufacturer.
• Look for privately run workshops for educators.
• Subscribe to computer education and Logo journals.
• Attend local, regional, and national computer education conferences.
• Sign up for a distance learning course in Logo.
• Locate area user groups, especially those that focus on Logo.
• Meet with other teachers in your district who are using Logo.
• Start your own Logo user's group.

If you have a modem, get online and seek out other Logo users. The Logo Foundation, in cooperation with the Global SchoolNet Foundation, manages a Logo discussion group available to anyone with access to Internet E-mail. The objectives of this Logo discussion group are to:

• Promote active sharing of ideas among Logo-using educators.
• Provide a forum for questions to be asked about using Logo in the classroom. This could be of service to both experienced and novice Logo educators.
• Provide a meeting place for Logo educators interested in collaborating on telecomputing projects involving their students sharing Logo creations.
• Provide a forum for discussion of educational philosophy on the user of technology in education and how Logo fits into that philosophy.

To join this Logo listserv discussion group, send an E-mail message to: majordomo@gsn.org
with the only line in the message: subscribe Logo-L

Books: (Some books may be out-of print, but are worth looking for.)

About Logo (or that contain major sections about Logo)
Bork, Alfred. Personal Computers for Education. Harper & Row, Publishers; New York, NY, 1985.
Harper, Dennis. Logo Theory and Practice. Brooks-Cole Publishing Company; Pacific Grove, CA, 1989.
Maddax, C. D. Logo in the Schools. Haworth Pr.; Redding, CA, 1985.
Muir, Michael. Fantastic Journey Through Minds and Machines. ISTE; Eugene, OR, 1991.
Papert, Seymour. Mindstorms: Children, Computers and Powerful Ideas. Basic Books, Inc.; New York, NY, 1980.
Papert, Seymour. Children's Machine: Rethinking School in the Age of the Computer. Basic Books, Inc.; New York, NY, 1993.
Papert, Seymour. The Connected Family: Bridging the Digital Generation Gap. Longstreet Press; New York, NY, 1996.
Solomon, Cynthia. Computer Environments for Children, The MIT Press; Cambridge, MA, 1986.
Turkle, Sherry. The Second Self: Computers and the Human Spirit. Simon and Schuester; New York, NY, 1984.
Weir, Sylvia. Cultivating Minds: A Logo Casebook. Harper & Row, Publishers; New York, NY, 1987.

Learning and teaching Logo
Beardon, D., Martin, K., and Muller, J. The Turtle's Sourcebook. Reston Publishing Company, Inc.; Reston, VA, 1983.
Bull, G., Tipps, S., and Riordon, T. R. Nudges: Apple Logo Projects. Holt, Rinehart and Winston; New York, NY, 1985.
Charishak, Ihor. Creating Dynamic Stories with LogoWriter. Dynamic Classroom Press; White Plains, NY, 1988.
Goldenberg, E. P. and Feurzeig, W. Exploring Language with Logo. The MIT Press; 1987.
Shimabukuro, Gina. Thinking in Logo: A Sourcebook for Teachers of Primary Students. Addison-Wesley; Menlo Park, CA, 1987.
Tipps, S. and Bull, G. Beginning with Logo: Terrapin Version. Prentice-Hall; Englewood Cliffs, NJ, 1987.
Watt, Daniel. Learning with Logo. McGraw-Hill Book Company; New York, NY, 1983.
Watt, M. and Watt, D. Teaching with Logo: Building Blocks for Learning. Addison-Wesley Publishing Co.; Menlo Park, CA, 1986.
Yoder, S. and Moursund, D. Logo PLUS (or LogoWriter) for Educators: A Problem Solving Approach. ISTE; Eugene, OR, 1990.

Logo as a programming language
Abelson, H. Logo for the Apple II. Byte/McGraw Hill; Peterborough, NH, 1982.
Anderson-Freed, S. and Brown, L. The Well-Tempered Turtle. Harvard Associates, Inc.; Cambridge, MA, 1995.
Friendly, Michael. Advanced Logo. Lawrence Erlbaum Associates, Publishers; Hillsdale, NJ, 1988.
Harvey, Brian. Computer Science Logo Style (in three volumes), The MIT Press; Cambridge, MA, 1985-87, 1996.
Spezeski, William. Logo: Models and Methods for Problem Solving. Harvard Associates, Inc.; Cambridge, MA, 1996.
Yoder, Sharon. An Introduction to Programming in Logo (Logo PLUS, Terrapin Logo/Macintosh). ISTE; Eugene, OR, 1988-1990.

Logo and Mathematics (see also Logo curriculum materials at end of Logo Directory)
Abelson, H. and diSessa, A. Turtle Geometry. The MIT Press; 1980.
Baker, J. and Hulme, D. The Expert Mathematician. New Directions Computer Mathematics Curriculum Development; 2018 James Ave. N., Minneapolis, MN, 55411, 1992.
Clayson, James. Visual Modeling with Logo. The MIT Press; 1988.
Cuoco, Al. Investigations in Algebra. The MIT Press; 1990.
Feurzeig, W. The Logo Language: Learning Mathematics Through Programming. Entelek, Inc.; Portsmouth, NH, 1977.
Hoyles, C. and Sutherland, R. Logo Mathematics in the Classroom. Routledge, Chapman and Hall, New York, NY, 1989.
Lewis, Phil. Approaching PreCalculus Discreetly. MIT Press; 1990.
Neufeld, Rudy. Learning Math with Logo, 1986. Learning Math with LogoWriter, 1990. Logo Publications; London, Ontario.

Textbooks featuring Logo
Billstein, Libeskind and Lott. Mathematics for Elementary School Teachers, 3rd edition. The Benjamin/Cummings Publishing Co., Inc.; Menlo Park, CA, 1987.
Clements, Doug. Computers in Elementary Mathematics Education. Prentice-Hall; Englewood Cliffs, NJ, 1989.
Cummins, J. J., Kenney, M., and Kanold, T. D. Informal Geometry. Merrill Publishing Company; Columbus, OH, 1988.
Serra, Michael. Discovering Geometry. Key Curriculum Press; Berkeley, CA, 1989.

Periodicals, journals, and magazines:
Logo journals or journals with a regular Logo section
The Computing Teacher, journal of the International Society for Technology in Education; 1787 Agate Street, Eugene, OR, 97403.
Logo Exchange, journal of the Special Interest Group for Logo-Using Educators, published 4 times a year by ISTE (see address above).
Computer education magazines with occasional Logo articles
Technology and Learning, 2169 Francisco Blvd. East, Suite A4, San Rafael, CA 94901.
Teaching and Computers, contact P.O. Box 2040, Mahopac, NY 10541.

Logo research:
Listing of available Logo Memos (pre-1986) and E&L Memos (since 1986). Epistemology and Learning Group, MIT, 20 Ames St. Room E15-309, Cambridge, MA 02139.
See also Logo Foundation (address below).

National organizations and Logo user groups:
CLIME (Council for Logo In Mathematics Education), 10 Bogert Avenue, White Plains, NY 10606
ISTE (International Society for Technology in Education), 1787 Agate Street, Eugene, OR 97403;
(503-346-4414); ISTE@uoregon.oregon.edu
Logo Foundation, 250 West 85th St., Suite 4D, New York, NY 10024 (212-579-8028)
SIG Logo, the Logo User's Group of ISTE (see address above).

Logo courses:
Independent Study Course offered by ISTE (see Journals for address): Introduction to Programming in Logo.
Graduate Logo courses offered by Lesley College, Cambridge, MA. Call 1-800-832-4808 for course offerings or to arrange a course.

Logo programming contest:
International Computer Problem Solving Contest. For information, contact: ICPSC, P.O. Box 085664, Racine, WI 53408.

Logo curriculum materials from Terrapin (see also our complete on-line catalog):
Birch, Alison. Logo Probability, 1988.
Birch, Alison. The Logo Project Book, 1986.
Cory, S. and Walker, M. Logo Works: Lessons in Logo, 1985.
Fitch, Dorothy. 101 Ideas for Logo, 1993.
Fitch, D. and Flanagan, P. Kinderlogo, 1984-95.
Picciotto, Henri. Logo Math: Tools and Games, 1990.