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   The computer serves as a powerful tool to assist students' learning and teachers' efforts to help learners to understand.  In my early years of teaching, microcomputers were not yet available.  When I began my career as a college professor, Apple II computers were becoming available.  I taught some early computer literacy classes for teachers.
   One of my favorite activities then was programming in Logo.  Seymour Papert, a professor at MIT, studied with Piaget, and used the cognitive development ideas he learned to create an environment in which children could experiment with mathematical ideas in much the same way that they engage naturally with language play.  Here is a sample picture that can be created using Logo.  An influential early book on computers in education is called The computer in school: Tutor, tool, tutee.  (For reference, see http://www.citejournal.org/vol3/iss2/seminal/article1.cfm   The idea is that in education, we may use the computer three different ways.  First, it can be a tutor (the computer teaches things to the learner).  The second use is as a tool (the computer helps us to do useful tasks like word processing or spreadsheets).  But the truly unique way is the third: the computer is the tutee - the learner teaches the computer.  In that case the learner grows through the process of teaching.
   Teachers should be careful to use computers for valid instructional goals.  There is some danger in teachers and students using computer activities that serve as little more than entertainment.  In the early years of instructional computing, many programs that were little more than "page turners" appeared.  These amounted to electronic delivery of content that could as well be presented in a book.  That is a waste of the computer's potential - and quite possibly the learners' time.
   There are some excellent programs that facilitate learning in ways generally not otherwise possible.  Probably the most famouse example of a simulation is Oregon Trail.  This allowed students to gain some experience of what it would have been like to travel with pioneer settlers from the midwest to Oregon in the 1800s in a covered wagon.  Obviously, a real experience like that would be nearly impossible for most teachers to arrange.
   One of the most important concepts in understanding learning is that of engaged time.  For learners to acquire meaningful understanding of a subject they must spend time actively engaged in studying and working with the content.  The Oregon Trail simulation prompted students to be more deeply engaged in studying the settlement of the American West than they were otherwise likely to be.  A Lemonade Stand simulation similarly engaged young children in economics and the math involved in selling drinks from a roadside stand.
   In my area of mathematics, a program I have become skilled in using is Geometer's Sketchpad.  This program is called "dynamic geometry."  Students can construct geometric figures and actively alter dimensions while observing the effects on the constructions.  For example, students can construct an arbitrary quadrilateral, bisect each of the 4 sides, and connect those midpoints to form a new quadrilateral.  Here are two sketches of doing so.  The amazing thing that students discover is that the interior quadrilateral will always be a parallelogram.
  One of the most common uses of computers in classrooms today is presentation software.  While not the only option for this, Power Point is one of the best know and most widely used.  As an aid to note-taking, PP presentations can be helpful to learners.  When posted on a classroom management system or a web site, they can be useful review tools for learners.  I am most attracted to uses that help draw students' attention to concepts and skills that may be challenging to grasp.  I am writing a textbook for a class called "Mathematics Comprehension for Teachers" that makes use of a variety of computer tools.  Long division is a topic that challenges many children.  Because many children make mistakes using the standard algorithm, teachers may find it helpful to have an alternative approach.  Here is a PDF version of a Power Point presentation that highlights the meaning of the "hang 7" algorithm that seems to be easier for many students to use.