A microworld is a term coined at the MIT Media Lab Learning and Common Sense Group (http://www.media.mit.edu). The lab defined a microworld as, literally, a tiny world inside which a student can explore alternatives, test hypotheses, and discover facts that are true about that world (www.umcs.maine.edu/~larry/microworlds/ microworld.html). It differs from a simulation (see Chapter 12) in that the student is encouraged to think about it as a “real” world, and not simply as a simulation of another world (for example, the one in which we physically move about in). Can you identify some mathematical related microworlds? (Also see Chapter 12.)


As noted earlier, an important part of computer literacy is to familiarize students with the process of programming computers to perform specific functions. In the 1960s, Seymour Papert (Papert, 1980), a mathematician, worked with a team to cre- ate the first version of Logo, which was successful in 1967. The Logo foundation (http://el.media.mit.edu/logo-foundation) is an excellent source of information about Logo. With the advent of Logo, programming skills can be taught to even the youngest students. The use of computer programming need no longer be limited to specialized secondary courses.

Many people have found that they can learn a great deal about programming by simply typing into the computer a Logo program that has been written by another person. This technique can be quite effective in the classroom. The student gains a sense of control over the computer by entering a set of program instructions and then observing the computer carrying out the instructions. Students come to understand some of the fundamentals in this manner. For example, they see that the computer can do only that which it has been programmed to do. They see the need for very accurate and concise instructions to avoid errors. They begin to understand that the computer can work for them once they learn to communicate with it.

Robotic clubs can help develop a basic understanding of creating, program- ming and building competitive robots. School robotic clubs are becoming very pop- ular, with local, state, and national competitions. Davis (2005) reported how Carl Hayden High School started a robotics club and won a national competition (see www.wired.com/wired/archive/13.04/robot.html). Lego has a robotics competition (www.usfirst.org/jrobtcs/flego.htm).

To begin, it is a good idea to begin exploring programming with user-friendly languages such as Logo. Students need to see that the computer can understand commands in plain English and that programming languages are not overwhelm- ingly complicated sets of numbers and esoteric symbols. Designing activities using Logo as the programming language is a good starting place because Logo is rela- tively accessible and understandable to students. Other options are Microworlds, Geo Logo, or LEGO MINDSTORMS Education. Carol (2006) provides an application for Lego using Lego pieces.

LEGO Mindstorms Education NXT has recently announced the next generation of educational robotics. This activity combines LEGO Education robotics and the lat- est technologies to provide learning activities for students ages 8 and up. Some of the features of the new system include:

  • New 32-bit NXT intelligent brick complete with a rechargeable battery system
  • Sturdy LEGO TECHNIC building system with more than 400 elements
  • New ultrasonic and sound sensors, plus improved light and touch sensors
  • Three interactive servo motors with built-in rotation sensors
  • Bluetooth technology allowing robots to communicate with computers, cell phones, and PDAs
  • New, highly intuitive programming software with increased functionality that, like ROBOLAB, is powered by LabVIEW™ from National Instruments
  • Curriculum, activities, and resources developed at Carnegie Mellon University

A number of books have been published with public domain programs that stu- dents can key into the computer for practice. Some of the programs perform basic calculations to solve problems or track numerical data, and others play games. The programs range from very simple to very complex, providing practice for students of all grade and skill levels. A word of caution is necessary, however. Not all the pro- grams to be found have been debugged carefully. Students may be frustrated if they spend hours typing and correcting their typed instructions only to find that the pro- gram does not work as promised even when the instructions are typed correctly. Teachers who possess strong programming skills can help their students out of such difficulties. Teachers with less experience in this area may do well to read reviews of the materials before using them in the classroom.

Another good source of programs for students to use is popular Logo computer periodicals. Some of these periodicals feature program instructions (also called code) and documentation on a regular basis. Many school libraries subscribe to these pe- riodicals, so the materials are often readily available to teachers and students.

The following Logo activity focuses on the area of mathematics. Gifted or highly motivated students can be encouraged to venture out on their own by tailoring the programs to perform additional functions as well. The activity focuses on using the computer as a tool for problem solving, which is really the forte of the computer. The authors suggest that teachers introduce programming early and in a nonthreatening way by integrating programming skills into subject areas across the curriculum. Specifically, because the computer is an excellent mathematical tool, it makes good sense to integrate the use of Logo to teach and explore mathematics. Students can learn programming skills with relative ease while they are mastering mathematical concepts. Teachers who want to introduce Logo to explore mathematics in their class- rooms can use the following activity.

Teachers should use their own judgment in adapting the activity for their grade levels and use with their own students. How well students perform in programming depends on a number of factors, including previous experience, attitudes toward computers, and perceived value of programming skills. Class discussions conducted before programming activities are presented can aid the teacher in discovering and exploring these crucial factors.


Activity Title: Problem Solving with Logo

Grade-Level Range: 4 to 6

Purpose.   In this activity students

  • Use Logo to solve mathematical and geometrical problems
  • Explore the concept of symmetry
  • Address the following National Educational Technology Standards (NETS) for Students 3—Technology productivity tools and 6—Technology problem-solving and decision-making tools

Activity  Preparation

  • Students need to be familiar with Logo commands used to draw
  • Prepare and copy activity
  • Draw several simple shapes on Then using the mirror, introduce the stu- dents to the concept of mirror images.

Activity Procedure

  1. Have students write Logo programs that will produce mirror images of the shapes displayed on the activity
  2. Students test their programs to see how well they
  3. Assist students in performing any debugging that may be required in order for the programs to re-create the shapes

Tools and Resources

  • Computer
  • Compatible Logo software
  • Activity sheet
  • Mirror

Assessment. Do the students’ programs draw mirror images of the shapes on the activity sheet? Have students been successful in locating and correcting errors in their programs?

Comments. Another interesting experience would be to give students only one-half of a shape so that the mirror images they create would complete the picture. Simple shapes such as houses or faces can be used effectively.