In the spring of 1990, I obtained a copy of Exploring Small Groups (ESG) (Geissinger, 1989 ) to use for demonstration purposes in my abstract algebra course. One day, as I was in the midst of explaining the commutator subgroup to my class, I stopped and sent the students to the computer lab to try to come up with some conjectures. The next day, they brought their conjectures to class. I picked three likely looking ones and asked my students to prove or disprove them for homework. One student raised her hand and said, "How can we do this? We don't know whether they are true or not?" I realized then that there was something terribly wrong -- not with the student, but with how I was teaching the material. This episode started me on a decade of experimenting with technology in my upper-level mathematics classes.
After my initial experience using ESG for abstract algebra, I dramatically changed how I taught that class. A weekly computer lab provided students with opportunities to easily construct some of the basic structures of group theory, such as subgroups, factor groups, and endomorphisms. The labs were discovery-oriented and always began with some paper-and-pencil work. The students worked in pairs in the lab and recorded their data during the lab session. One of my favorite labs takes advantage of ESG’s use of color to illustrate quotient groups. [The "lab" link in the preceding sentence will show you the lab in a browser window. To download the original Scientific Notebook file (a .tex file), click on the icon at the right.] For example, if one chooses the group 0804 (the symmetries of the square, ) from the group library of ESG, generates the center of that group, and then forms the quotient group of , one obtains the colorful image shown here.
By the time my students are working on this lab, they are familiar with the patterns of most of the groups of low order. So it is easy for them to recognize that this colorful table gives the pattern of .
I have always believed that the true learning takes place when the students write up their lab reports, and at the end of the reports they were asked to make conjectures. Their conjectures sometimes anticipated material in the text, often with different phrasing and notation. [Again, the "conjectures" link will open in a browser window. To download the original Scientific Notebook file, click on the icon at the right.] It was especially nice when a conjecture gave me the basis for a new classroom discussion. Follow-up sessions emphasized the theory and proofs that are so fundamental to this course. Students were also expected to write formal mathematical proofs for this class, in addition to the less formal, expository paragraphs in their lab reports. The lab materials have been published by the Mathematical Association of America in the lab manual, Laboratory Experiences in Group Theory (Parker, 1996 ).
Why does discovery seem so natural in this course? A major factor in my class was the excellent software, Exploring Small Groups, which is extremely easy to use. However, other faculty are successfully using a variety of software -- ISETL (Dubinsky and Leon, 1993 ), Mathematica (Wolfram, 1988-2000 ), Geometer's Sketchpad (Jackiw, 1992-95 ), for example -- to teach this course. The key is that the subject matter is new to virtually all students. Therefore, the underlying goal of the course is to introduce the basic concepts or constructions. Students develop an intuitive sense of the concepts as they investigate examples generated by hand and by the software. And, as they move through the semester, they formalize the concepts into the standard language of mathematics. Conjecture-posing is an excellent way for students to play with the ideas. In some sense, it doesn't matter what the statement of the conjecture actually is. To make a conjecture, the students need some kind of understanding about the nature of that mathematical object. In an introductory algebra course, they are primarily learning the basic structures rather than a long list of theorems. In my experience, the conceptual understanding and the process of formalizing these concepts are done simultaneously in an abstract algebra course, and technology is an aid in the process.