Effective Teaching of Abstract Algebra
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Abstract Algebra is one of the important bodies of knowledge that the mathematically educated person should know at least at the introductory level. Indeed, a degree in mathematics always contains a course covering these concepts. Unfortunately, abstract algebra is also seen as an extremely difficult body of knowledge to learn since it is so abstract. Leron and Dubinsky, in their paper ¡§An Abstract Algebra Story¡¨, penned the following two statements, summarizing comments that are often heard from both teacher and student alike.
1."The teaching of abstract algebra is a disaster, and this remains true almost independently of the quality of the lectures." (Leron and Dubinsky agree with this statement.)
2."There's little the conscientious math professor can do about it. The stuff is simply too hard for most students. Students are not well-prepared and they are unwilling to make the effort to learn this very difficult material." (Leron and Dubinsky disagree with this statement.)(Leron and Dubinsky, p. 227)
Thus the question is raised: if there is something the "conscientious math professor can do about" the seemingly disastrous results in the learning of algebra, what is it that we can do? As a teacher of undergraduate mathematics, I want and need to know what these effective methods of teaching abstract algebra are.
Leron and Dubinsky's paper referred to above and papers resulting from their research contain the bulk of literature that I reviewed. In this paper, they summarize their
experimental, constructivist approach to teaching abstract algebra. Among the classroom activities are computer activities, work in teams, individual work, class discussion, and sometimes a mini-lecture summarizing the results of student work (which by this time is familiar to them), providing definitions, theorems, and proofs in their abstract forms.
The computer activities use the ISETL programming language. As an example of its use, students write a program implementing the group axioms. They then can enter what they consider to be a group, and the computer will give as output a true or false response. They can use the same process to determine whether their proposed group is closed, has an identity, etc. They choose their answer and then let the computer respond. In this way, students ¡§construct¡¨ the group process, with the view that they will also have a ¡§parallel construction¡¨ occurring in their minds. Students have an experience on which to base their learning of group theory.
The method proposed here by Leron and Dubinsky certainly seems patterned after Dubinsky's theoretical foundation for student learning laid out in his work ¡§Reflective Abstraction In Advanced Mathematical Thinking.
¡¨ The construction of schema in this paper outlines the construction of the group process that occurs here. Leron and Dubinsky state, ¡§. . . knowledge is not transferred from one person (the teacher) to another (the student), but rather is constructed in the student¡¦s mind.¡¨ His APOS theory is also referred to, for example, as he discusses the process of forming a coset, and the difficulty of the student to see the coset as an object that can then be acted upon.
Leron and Dubinsky claim, ¡§replacing the lecture method with constructive, interactive methods involving computer activities and cooperative learning can change
radically the amount of meaningful learning achieved by average students.¡¨(Leron and Dubinsky, p. 227)
Is their claim justified? Leron and Dubinsky say, ¡§Experience, theory and research all point to the fact that verbal explanations that do not relate to the student¡¦s prior experience are quite ineffective.¡¨(Leron and Dubinsky, p. 231) They have found that there are, of course, problems that need to be worked out, but the ¡§revolution!¡¨4 in student learning makes it worthwhile. One example of a topic of abstract algebra in which they specifically say that their method improves student learning is that of the quotient group. Their claim, backed up by their research, is that ¡§students don¡¦t feel the same alienation and paralysis in the face of the quotient group.¡¨ (Leron and Dubinsky, p. 240)
What about others in the field? Do they propose similar methods? Joseph Gallian, in fact, proposes a similar tactic in connection with his textbook Contemporary Abstract Algebra. The text incorporates Java applets and the use of GAP, free software that yields data concerning groups and rings. He writes, ¡§This software (Java) allows students to easily produce data to make and test conjectures.¡¨ These appear to allow the instructor to employ this method that encourages the construction of group (and ring) processes so that the concepts are familiar before viewed in their abstract forms. I have not read research relating to his proposed use of these or of his proposed method, however.
Dubinsky¡¦s method was specifically tested at Middle Tennessee State University after faculty heard his presentation at the 1992 International Conference on Technology in Collegiate Mathematics. The undergraduate mathematics program at Middle Tennessee State was being restructured, and in the fall of 1993, as part of this restructuring, use of
ISETL was implemented in the abstract algebra course as an opportunity to try Dubinsky¡¦s method. Vatsala Krishnamani writes, ¡§These changes resulted in student discoveries. Computer activities enabled the teacher to introduce concepts well in advance of when they could otherwise have been introduced and enabled the students to be more familiar with the ideas, abstract and otherwise. The activities of the students also reflected changes. They diverted from ¡¥studying just for tests¡¦ to a pattern of continuous learning and having discussions with peers and professors. They became comfortable with making predictions. Some took the role of teachers in their groups as well as in the entire class. Retention improved. Students were proud to present their discoveries.¡¨(Krishnamani, p. 2) Vatsala goes on to say that student understanding of specific group concepts improved vastly when compared to results using traditional teaching methods. This understanding, in fact, went beyond the usual expectations for the course. The only problems reported were in terms of initial student anxieties about learning using an unfamiliar method.
A third paper I reviewed is entitled ¡§Coordinating Visual and Analytic Strategies: A Study of Student¡¦s Understanding of the Group D4¡¨, by Zazkis, Dubinsky, and Dautermann. This study was investigating the ¡§interrelationship between visualization and analysis in terms of student learning.¡¨ This, too, stems from Leron and Dubinsky¡¦s paper on teaching/learning abstract algebra, as students from two midwest universities were interviewed after completing an abstract algebra course taught using computer activities and cooperative learning strategies described by Leron and Dubinksy.
In the group D4, a visual approach to group operations is considered to be making a square, labeling the corners, and performing the reflections and rotations that represent
the group operations of the symmetries of a square. An analytic approach is multiplying four objects according to an algorithm, the result being a permutation product. Detailed definitions of visualization and analysis are given in the paper, but are somewhat lengthy. Leaving out many of the details, visualization is described as doing something with a physical or mental object, but if done to a mental object, it must have reference to an external object. Analytic thinking is described as mentally manipulating objects or processes, without this external object to refer to.
In the interviews, students could not be categorized as visual thinkers or as analytic thinkers, as all students had a mixture of both. The conclusion of the research is that both types of thinking are required, and the writers describe a model called the ¡§Visualization/Analysis model.¡¨ The model can be depicted by a triangle with levels of visualization going up one side and levels of analysis going up the other. There are paths that lead around the triangle. At V1, the student constructs a process, and as he becomes familiar with it, he is led to consider it in an analytical way, A1. The student is led to take another look and reevaluate the process visually, V2, and then to consider this analytically, A2, and so on, until the two ways of thinking become closer and understanding is reached. In the beginning, the two ways of thinking were very different and hard to negotiate between, but as the model is applied, the two become so similar that they blend together, and interiorization occurs.
To succeed in abstract algebra, one must eventually be able to accomplish several things. One must be able to visualize algebraic objects and their properties, as well as
analyze them. One must be able to think analytically, with the ability to understand and apply proofs, and also to write them.
Since I will be new at teaching abstract algebra, my only experiences with it deal with my own learning, which, until recently, has all been in the traditional way. While most of my earlier professors gave many examples and urged my classmates and I to refer to them throughout the course, the courses were taught by the lecture method. Any construction of objects had to be done on our own. Anything that I felt I fully understood, I somehow constructed in my mind before true understanding took place.
I can see the validity of the method proposed by Leron and Dubinsky. Construction of objects in algebra must occur, but I don¡¦t feel that it must occur via the computer activities, as long as it happens. Piaget¡¦s Reflective Abstraction directly applies here, along with Dubinsky¡¦s APOS theory, as we consider actions and processes on groups, and then groups, subgroups, etc., becoming objects to be acted upon. The Visual/Analytic Model can be used to show specifically how the learning can take place. Within this model, reflective abstraction, APOS, and even my earlier model, that of combining problem solving and Van Hiele levels can be seen at work. A specific example using the Visual/Analytic Model is seen in figure 1. After an explanation of how the model works, we can see more clearly how the above principles apply.
Most students in an introductory abstract algebra course have no idea what a group is. At the bottom left of the triangle in figure 1, students are directed to start by visualizing some groups. A discussion of clock arithmetic and operations in Z5, as well as symmetries of the square and operations in D4 should take place so students have an initial knowledge base from which to begin. For visualization to take place, form groups
Any groups inside ƒ¤Define a subgroup
Are the following groups? Are they associative? _-w Formal Definition of a group
What properties do Informal Definition of a group and have students make Cayley tables. Students are directed to look for patterns and to discuss ways the operations could be performed analytically. After the discussion, they summarize their findings in a more formal way. In doing so, students have been led from the visual to the analytical. Some construction should be taking place in their minds concerning what this object called a group is.
To build on this construction, discuss properties that Z5 and D4 have in common. To give students a starting place, first discuss the properties of the real numbers. Students should already know these, but may need a reminder. Students, in their groups, will need to perform operations to determine if inverses and identities are present, and should check to see what properties of the real numbers apply to these two groups. Once discussions have finished, an informal definition of a group can be made. Thus, students have crossed the triangle once more from visual to analytic thinking, and further construction of groups has taken place.
Again, build on this construction by presenting sets with a binary operation, some of which are groups and some of which are not. For instance, give students the groups Z under addition, Z under multiplication, or matrices under addition or multiplication. Students may come up with their own examples. Students should check to see if these sets with the given operation are associative, and if they have identities and inverses. We want to turn to analysis once again, to a formal definition of a group. To do this, a common notation would be helpful. Groups can discuss what such a common notation might be before a class discussion and a formal definition is given. Once more, the construction of groups in the minds of students has occurred. Notice that visual and
analytic thought is becoming closer and it is not quite as difficult to make the link between the two.
Build on this construction again. Student groups have already determined that groups have identities and inverses. How many of each does a single group have? Does a group have the cancellation property? Students should examine their groups, performing the group operations to determine these things for particular groups. Ask the student groups what statements we could make and how we could prove that these statements are true for all groups. They will need to reference group notation previously determined. At this time, the concept of a theorem and its proof should be discussed as a class. The proofs for uniqueness of the identity and the inverse should be shown formally, as well as the proof for the cancellation property. Again, students have been brought from visualization to analysis.
This process continues until at the top of the triangle, students understand groups from a combination of visual and analytic thinking, and groups and subgroups have now become objects that are acted upon.
Piaget¡¦s reflection is apparent everywhere in this model of teaching abstract algebra. At each step in the model, actions are performed on objects until we have new objects to act upon. Construction of a mental process relating to the group process helps students to interiorize the process. Coordination occurs, for instance, when groups are found inside of groups. The result is encapsulated into the object subgroup. Generalization occurs to define groups, subgroups, and theorems relating to them. A schema is formed that can be applied to all groups. Students have constructed their understanding of groups and subgroups. The reversal of this construction would be to then apply, for instance, the definition of a subgroup to a set with a binary operation to determine if it is a group.
At every level of abstract algebra, this construction can occur, and must occur if one is to understand it fully. The breadth of knowledge of abstract algebra is too large to learn by memorization alone. Rather, the understanding of how things work at every level is necessary. This understanding comes from the construction just described.
Dubinsky¡¦s APOS theory also describes the construction seen in the model. Active construction occurs on objects that students are knowledgeable about, thus bringing the motivation to learn. At every step of the model, processes are applied to objects, until new objects are recognized. As generalization occurs to include all groups, rings, fields, etc., schemas are formed on which students can apply. The APOS theory can be applied to every topic in abstract algebra and at every level.
The incorporation of Van Hiele levels into the problem solving process, seen in my first paper, also applies to learning abstract algebra. At the visualization side of the triangle, questions are posed, the problem to be solved. Students may explore and design several methods of solution to the problem before the question posed is finally answered. In doing so, they are going through the problem solving process. The existence of Van Hiele¡¦s visual level is obvious, as the entire visualization side of the triangle is on this level. The descriptive level occurs when students can summarize and describe their findings. We reach the theoretical level on the analytic side of the triangle, when definitions and theorems are understood and stated. The fourth level, that of formal logic occurs when theorems are proved. When students can then write their own proofs, the fifth level, the nature of logical laws, is reached.
I concur with the common thought that the lecture method of teaching abstract algebra is not successful with most students, at least at the undergraduate level. The act of constructing groups, etc., makes sense, and research shows that constructivist methods have been successful for student learning in abstract algebra. Having had no experience in teaching abstract algebra myself, but looking at teaching the course in the near future, it would be prudent to try methods that have been successful. For the moment, I would have to say that this method of teaching appears to be the best method to try. It will be when I teach the course and have an experiential base of my own that I will truly be able to say whether I believe or do not believe that these methods are the best for student understanding of abstract algebra.
Dubinsky, Ed (1991), ¡§Reflective Abstraction In Advanced Mathematical Thinking,¡¨ in
Advanced Mathematical Thinking, Tall, David (ed.), pp. 95-123, Kluwer
Academic Publishers, Boston, MA.
Gallian, Joseph (2002), Contemporary Abstract Algebra, 5th Edition, Houghton Mifflin
Company, New York, NY
Krishnamani, Vatsala and Dovie Kimmins (2001), ¡§Using Technology as a Tool in
Abstract Algebra and Calculus: The MTSU Experience,¡¨ paper presented at
the Annual Meeting of the International Conference on Technology in Collegiate
Leron, Uri and Ed Dubinsky (1995), ¡§An Abstract Algebra Story,¡¨ The American
Mathematical Monthly, v. 102, No. 3, 227-242
Van Hiele, Pierre (1986), Structure and Insight:A Theory of Mathematics Education,
Academic Press, New York, NY
Zazkis, Rina, Ed Dubinsky, and Jennie Dautermann (1996), ¡§Coordinating Visual and
Analytic Strategies: A Study of Students¡¦ Understanding of the Group D4,¡¨
Journal forResearch in Mathematics Education, v. 27, No. 4, 435-437