Technology in the Upper-Level Curriculum

Author(s): 
Ellen J. Maycock

Many of us who teach mathematics have been using technology in our classrooms for over a decade. Having a computer available on a daily basis has become standard for me -- in fact, I feel bereft if I enter an assigned classroom a few days before the beginning of the semester and discover that there is no computer in the room. Our students, after learning calculus with a laboratory component, have moved on to upper-level courses with new expectations about the learning environment. We knew there would be substantive differences, compared to calculus, in the way computers would be used in theoretical courses.

Ellen Maycock is Professor of Mathematics at DePauw University.

However, many of my colleagues and I anticipated that there would be common themes in the uses of technology in upper-level classes. Indeed, two conferences -- the FIPSE-supported Conference on Technology in the Upper-Level Curriculum, held at St. Olaf College (1994), and a subsequent NSF-UFE conference, Exploring Undergraduate Algebra and Geometry with Technology, held at DePauw University (1996) -- were based on the idea that there would be a fundamental similarity in the use of technology in these upper-level, axiomatic courses.

I had, in the early and mid-1990’s, developed laboratory materials to teach abstract algebra with a computer laboratory component (Parker, 1995, 1996 ). Additionally, as part of a Mellon Foundation grant to DePauw, I was chosen to be one of a pilot group of faculty to introduce technology into a course that had been previously taught without any. I decided to experiment with our department’s introductory real analysis course in the spring of 1999, and I confidently planned my syllabus assuming that I could use my successful pedagogical techniques from abstract algebra in real analysis. Because of this underlying assumption, I was unprepared for the problems that I encountered in this experimental real analysis course. After having candid discussions with students midway into the semester, I realized that there were subtle but distinct differences in the nature of our department’s upper-level courses that meant that technology would play a significantly different role in each.

I present here my experiences in three upper-level courses: abstract algebrareal analysis, and geometry. Certainly, there are recurring themes in the use of technology in all of these courses, and I can only repeat here what many others have also experienced in their classrooms. For me, the use of a laboratory component has been very successful in all three courses, but only after I understood that I could not automatically transfer the format from one course to another. I share these observations to encourage the reader to consider thoughtfully how technology can enhance the learning environment in each classroom. I would like to pose the bulleted conclusions on page 5 as research topics to be explored more fully, both in theoretical pedagogical discussions and in controlled classroom experiments.

Published May, 2002
© 2002 by Ellen J. Maycock

Technology in the Upper-Level Curriculum - Introduction

Author(s): 
Ellen J. Maycock

Many of us who teach mathematics have been using technology in our classrooms for over a decade. Having a computer available on a daily basis has become standard for me -- in fact, I feel bereft if I enter an assigned classroom a few days before the beginning of the semester and discover that there is no computer in the room. Our students, after learning calculus with a laboratory component, have moved on to upper-level courses with new expectations about the learning environment. We knew there would be substantive differences, compared to calculus, in the way computers would be used in theoretical courses.

Ellen Maycock is Professor of Mathematics at DePauw University.

However, many of my colleagues and I anticipated that there would be common themes in the uses of technology in upper-level classes. Indeed, two conferences -- the FIPSE-supported Conference on Technology in the Upper-Level Curriculum, held at St. Olaf College (1994), and a subsequent NSF-UFE conference, Exploring Undergraduate Algebra and Geometry with Technology, held at DePauw University (1996) -- were based on the idea that there would be a fundamental similarity in the use of technology in these upper-level, axiomatic courses.

I had, in the early and mid-1990’s, developed laboratory materials to teach abstract algebra with a computer laboratory component (Parker, 1995, 1996 ). Additionally, as part of a Mellon Foundation grant to DePauw, I was chosen to be one of a pilot group of faculty to introduce technology into a course that had been previously taught without any. I decided to experiment with our department’s introductory real analysis course in the spring of 1999, and I confidently planned my syllabus assuming that I could use my successful pedagogical techniques from abstract algebra in real analysis. Because of this underlying assumption, I was unprepared for the problems that I encountered in this experimental real analysis course. After having candid discussions with students midway into the semester, I realized that there were subtle but distinct differences in the nature of our department’s upper-level courses that meant that technology would play a significantly different role in each.

I present here my experiences in three upper-level courses: abstract algebrareal analysis, and geometry. Certainly, there are recurring themes in the use of technology in all of these courses, and I can only repeat here what many others have also experienced in their classrooms. For me, the use of a laboratory component has been very successful in all three courses, but only after I understood that I could not automatically transfer the format from one course to another. I share these observations to encourage the reader to consider thoughtfully how technology can enhance the learning environment in each classroom. I would like to pose the bulleted conclusions on page 5 as research topics to be explored more fully, both in theoretical pedagogical discussions and in controlled classroom experiments.

Published May, 2002
© 2002 by Ellen J. Maycock

Technology in the Upper-Level Curriculum - Abstract Algebra

Author(s): 
Ellen J. Maycock

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.

Technology in the Upper-Level Curriculum - Real Analysis

Author(s): 
Ellen J. Maycock

When I had taught real analysis previously, I had been frustrated by the students' lack of understanding of this very abstract material. So I was eager to see how technology could be used to help students grasp these difficult concepts. In the spring of 1999, I taught real analysis using the computer algebra system Mathematica. I planned the course around six or seven standard topics from real analysis and used a standard, formal text. Each topic was to be covered using the format of an introductory discovery session in the lab, classroom discussion with formalism introduced, and a final class session where students presented solutions to problems.

In the early part of the semester, the classroom sessions were challenging and enlightening for the students, but I was having limited success in connecting their intuitive ideas, developed in beginning calculus, to the formalism of the real analysis text. By mid-semester, it was clear that the discovery approach in the laboratories -- which had been so successful in abstract algebra -- was not working well. Fortunately, the students were comfortable articulating their frustrations in a long classroom discussion. Although the computer-generated examples enhanced their intuitive understanding of topics they had learned in calculus, they told me that they definitely had no idea of how this intuition related to the formalism that I had been emphasizing in the classroom.

During the next class, I wrote a simple   proof on the board, and I also projected the computer screen with the Mathematica information generated in their lab: the familiar graph showing the   and related , and a table of numerical values for 's and 's. In the formal example I used, the computations for the proof implied that   , and the numerical tables showed this as well. [To download the original Mathematica notebook, click on the icon to the right. The "example" link shows the same information in a browser window.] The formal proof, the graph, and the numerical tables were all in front of the students at once -- I had to move between these various presentations of the same concept several times. Finally, one by one, the students’ faces signaled understanding. They had grasped the meaning of the formal mathematical proof, but only after they had seen the visual and numerical illustrations of it.

After that point, I changed the construction of the remaining units. I had a careful presentation in the classroom of each new topic before we went to the lab. The lab then gave numerical and graphical support of the concept; the follow-up after the lab reinforced the connection between the concepts and the formalism. There were many "aha!" moments, in my office and in the classroom. One of the most successful labs illustrated the idea of uniform convergence, using the Taylor series of the sine function. [For the original Mathematica notebook, click on the icon at the right.] By substituting increasingly large values of n into the code, students were able to discover when the Taylor series for finally fit into the outline formed by the two curves and , for the interval  .

In this real analysis class, students could see how the formalism of the text was the final step in the process -- when the ideas were put into the language of mathematics. They especially enjoyed their own successes in writing proofs and presenting them to their classmates. They finally understood that the purpose of the course was to make more precise and formal some deep mathematical concepts, and that we were trying to investigate these concepts in a variety of ways.

By the end of the semester, in a classroom discussion that was taped, students expressed a very positive attitude about the course and the laboratory experience. They felt that they had struggled with some highly abstract material, but the varied activities of the course helped them understand the concepts deeply. The technology was one component -- but clearly not the only one -- that helped them learn.What we all came to realize is that the use of technology in this course helps students bridge the huge gap between their very loose grasp of the concepts learned in beginning calculus and the highly formal and somewhat non-intuitive rigor of real analysis. My contribution to the multidisciplinary DePauw Faculty Instructional Technology Support program (FITS) was twofold:  first, that the instructor cannot assume that techniques that work in one course will transfer to another course, and second, that the success of such a course can be enhanced by allowing the students to be part of the process.

 

Technology in the Upper-Level Curriculum - Geometry

Author(s): 
Ellen J. Maycock

Always ready to try technology, I decided to teach DePauw's junior-level geometry course with a variety of tools: Geometer's Sketchpad (Jackiw, 1992-95 ), the Lénárt Sphere (Lénárt, undated), and PoincaréDraw (The Gap Group, 2000  -- written by Robert Foote and Nathan of Wabash College to demonstrate the Poincaré disk model of hyperbolic geometry). I continually felt challenged and stimulated by these students -- I am pleased that atmosphere has been repeated during several subsequent semesters. The question I have asked myself about this course was, "Why did we all feel so energized by this course? Why did the software do so much for the students?"

My students had all studied Euclidean geometry thoroughly in high school. They had been taught how to write proofs for geometry, and it was easy for me to show them how to transform their 2-column proofs into paragraph form. I then sent them to the computer lab to see what Geometer's Sketchpad would do. What was truly exciting for them was the enhanced capacity for visualization that came from the dynamic software. This became clear when I insisted that they work out illustrations for one problem set by hand, with a straightedge and compass. They struggled with the hand sketches and invariably missed the "special cases" that made refinements of the conjectures necessary. Later, I had them work with the Lénárt Sphere, and they fairly quickly recreated some of the crisis of 19th century geometry. Finally, they learned about hyperbolic geometry and appreciated how much PoincaréDraw helped in visualizing a "different" geometry. In the lab "Inscribed and Circumscribed Circles" students work through the same constructions in all three geometries and compare the results. [For the original Scientific Notebook file, click on the icon at the right.]

Our modern students do not have much experience with visualization in earlier courses -- in either two or three dimensions -- and this limits their abilities to think about geometry. The dynamic software packages (Geometer’s Sketchpad and PoincaréDraw) enhanced their visualization and allowed them to "think gometrically." They had already understood the basic concepts and had been taught to formalize these concepts. But the new capacity to visualize, provided by the software, opened up creative avenues for them. As one student said at the end of the semester during a taped discussion, the course showed them the "wild side of math." More than in abstract algebra or real analysis, the students understood and were stimulated by the potential for mathematical exploration.

Technology in the Upper-Level Curriculum - Conclusions

Author(s): 
Ellen J. Maycock

It is clear that technology can enhance the learning environment in quite different ways. Observations I have made in my own classrooms, as well as student comments in their evaluations and in taped discussions, have led me to draw the following conclusions. These statements should be regarded, however, as conjectures that need further investigation by the mathematics profession as we move into classrooms that are more and more technologically sophisticated.

  • For new material, technology can help students master concepts and formalism simultaneously.
  • For material that was previously learned intuitively and informally, technology can bridge the gap between intuition and formalism.
  • For material that has been learned well at an elementary level, technology can help students explore more advanced material creatively and independently.

There are themes that ran through all the courses, and I can only repeat here what has been discussed at length in our profession. Technology allows students to investigate examples more easily and enhances visualization. Instead of working through one example with paper and pencil in the course of an hour, the student can generate six or eight with the computer -- and with the dynamic geometry software, thousands. Patterns can emerge from the examples. Students are much more able to see the concepts behind the formalism and the theory. 

Primarily, however, the lab experiences changed the dynamics of the courses. A carefully constructed syllabus became a hindrance for each course -- the unpredictability of the lab experience meant that I had to be prepared to discard my lesson plans for the day and respond to their comments and questions. I had to ask myself what my basic goal was in each class and be flexible about whether a list of theorems could be covered -- I refocused on a sparse collection of fundamental concepts in each course. Students felt empowered by their own discoveries, and they began to provide at least as much energy to the classroom as I did. The students were quick to discern that they had more control of the learning environment and responded to my pleasure in the change.

One of my classroom "triumphs" came during a problem set presentation in geometry. I had inadvertently written out a problem incorrectly for the class. The student who was to present it came to the board and said meekly, "I think this problem is not correct." When I said that on the surface it had seemed OK to me, she stomped her foot and shouted, "I know this is wrong!" She proceeded to show the class the counterexample she had constructed with Geometer’s Sketchpad . This indeed is success in the classroom!

Technology in the Upper-Level Curriculum - References

Author(s): 
Ellen J. Maycock

Dubinsky, E., and U. Leon (1993), Learning Abstract Algebra with ISETL, Springer-Verlag

Foote, R., and N. Fouts (1998), PoincaréDraw, Crawfordsville, IN

The GAP Group (2000), GAP--Groups, Algorithms and Programming, Version 4.2; Aachen , St. Andrews

Geissinger, L. (1989), Exploring Small Groups, now available only through (Parker, 1996)

Hibbard, A., and K. Levasseur (1999), Exploring Abstract Algebra with Mathematica,  TELOS/Springer-Verlag

Jackiw, N. (1992-95), The Geometer's Sketchpad, Key Curriculum Press

Lénárt, I. (undated), Lénárt Sphere, Key Curriculum Press

Parker, E. Maycock (1995), "A Laboratory Experience in Group Theory," UME Trends, vol. 6, no. 6, p.24

Parker, E. Maycock (1996), Laboratory Experiences in Group Theory, The Mathematical Association of America

Schattschneider, D. (1997), "Visualization of Group Theory Concepts with Dynamics Geometry Software," in Geometry Turned On, D. Schattschneider and J. King, eds., The Mathematical Association of America

Wolfram, S. (1988-2000), Mathematica, Wolfram Research, Inc.