Dancis, J. & Davidson, N. "The Texas Method and the Small Group Discovery Method." (1970).

*Jerome Dancis wrote a Ph.D. thesis in topology under the direction of RH Bing, a student of R.L. Moore. He is a professor of mathematics at the University of Maryland.*

*Neil Davidson studied topology at the University of Wisconsin and wrote a Ph.D. thesis in mathematics education on the small group discovery method. He is a professor education at the University of Maryland and past president of an international society devoted to cooperative learning.*

**INTRODUCTION**

We shall discuss two methods of instruction in which students function as junior mathematicians. The purpose of this paper is to describe these methods and their goals, in the hope of inducing some of our colleagues to try them.

The "basic thesis" of the "Texas" method or Moore method is that bright students can develop mathematics by working individually within a competitive system. The basic thesis of the small group discovery method is that average or better students can develop mathematics by working together as a small group within a cooperative system. We shall discuss these methods only in the context of teaching individual classes which are part of a standard curriculum.

The methods described here have special goals, namely: To directly help students develop some of the abilities which characterize the working mathematician. In general, these methods are used to develop mathematical creativity and intuition, as well as skill in logical reasoning. In particular, these methods concentrate explicitly on (1) the ability to invent techniques for solving problems and proving theorems, (2) the ability to make conjectures and educated guesses, and (3) the ability to present a coherent argument. The students develop these abilities as they master a moderate amount of mathematical knowledge.

With the methods described here, emphasis is placed on the discovery of new ideas by the students. The students learn mathematics as active participants, rather than as spectators. There are opportunities for students to present their ideas and results to their peers and to the teacher.

Basically, an instructor who is using these methods does not lecture; instead he provides his students with a good outline of the course. The students establish the theorems, solve the problems, and work out the examples in this outline. With these methods, the students spend considerably more time proving theorems, solving problems, making conjectures, and presenting their results than they do in the average mathematics course. With these activities, the students participate in building the mathematical theory in a course. Thus, students learn by direct experience that mathematical results are developed by human beings and are not handed down "from on high."

In the next two sections we shall present a partial description of the Moore method and of the small group discovery method. In the fourth and fifth sections we shall discuss mathematical content and procedural considerations which are common to both methods.

**THE TEXAS METHOD**

The Texas (or Moore) method was developed and used very successfully by Professor R.L. Moore for half a century at The University of Texas. (See [3, 4, 7, 8].) Many people consider the Texas method to be precisely R.L. Moore's style of teaching at The University of Texas. We take a much broader view, which can encompass the variations on the Texas method used by many teachers. Our discussion will concentrate upon basic procedures which can be used, in our opinion, by most practitioners of this system.

Moore developed his teaching method as a complete mathematical education system: From trigonometry and calculus through the Ph.D. thesis. He developed his own curriculum and course syllabi, and decided the class size, which was frequently quite small. He restricted class membership to students who agreed to cooperate with his methods. He selected his students on the basis of personal interviews or past performance in courses taught by his methods.

In contrast, here we will only discuss Moore's method in the context of teaching individual mathematical courses which are part of a curriculum. Thus we have more constraints and considerably less freedom than did R.L. Moore.

We now proceed to describe certain aspects of the Moore method. Under this method each student proves as many theorems and solves as many problems as possible by himself outside of class. For each theorem or problem, one student presents, at the blackboard, a proof or a solution that he devised by himself. Students are not permitted to discuss mathematics outside of class. The Texas method makes use of a competitive atmosphere. There is competition among students to impress the teacher and to solve more of the difficult problems than their classmates. Friendly competition spurs many students to work harder than they would otherwise, and occasionally to work above and beyond reasonable limits (i.e., to the detriment of other courses).

Students sometimes work on a difficult theorem for a week or more until someone finally proves it. In advanced courses some of the problems are open questions, and some of their solutions have become Ph.D. theses. Thus, the Texas method requires and tends to produce student willingness to work on problems for long periods of time. Considerable increases in a student's self confidence will result from his solving difficult problems.

We now present a brief description of blackboard procedures. In selecting someone to present a solution to the class, the teacher picks one of the students who claims to have *already* solved the problem by himself outside of class. Some teachers prefer to call on the weaker students first. (If a student is hot upon the trail of a theorem and doesn't want to see someone else's proof, he may leave class during the presentation.)

Suppose a student is presenting a proof to the class, and a mistake is found in the proof. If the student can repair the damage immediately - fine; however, the teacher does not permit the student to flounder at the blackboard while attempting to repair his proof. If the mistake in the student's proof can be corrected, the student is permitted to try again, usually the next day. However, if the student shows little understanding of the problem, the teacher might choose to send another student to the board to present a solution.

During a weak student's presentation, the instructor *should* be supportive. This includes helping the student clarify *his* [the student's] ideas and statements. Proper use of this system does *not* include the teacher embarrassing or actively discouraging the weaker student; he does not permit other class members to do so either.

One method of grading is for the instructor to give a standard-type final examination and to use the results of this examination as the main criteria for grading those students who have presented few results to the class.

The Texas method does a spectacular job of differentiating between the stronger and the weaker students. This method has been used as a filtering device to identify mathematical talent. The filter has also convinced other students that they do not really want to become mathematicians.

The Moore method is well suited for courses at the senior and first year graduate level and for some research courses. A senior level course in topology or complex variables is a good course for a teacher to first try out the Moore method. The authors have no experience and some apprehension about using the Moore method in isolated freshman or sophomore courses which are part of a standard curriculum.

We conclude by noting that the Moore method tends to develop ambition, a competitive spirit, and perhaps "Texas-style individualism." Its most striking success has been in the field of point-set topology. The list of R.L. Moore's doctoral students includes a sizable number of outstanding researchers in topology. [7]

**THE SMALL GROUP DISCOVER METHOD**

The small group discovery method [1, 2] was developed in 1967 at the University of Wisconsin by Neil Davidson. He started with the idea established by R.L. Moore that bright students can develop mathematics. He then changed the social environment so as to render the idea workable for much larger numbers of students in undergraduate courses.

Under the small group discovery method the class is divided into small groups with three or four members per group. Each group discusses the theorems (or problems) and proves them cooperatively as a group effort during class. Every group has its own section of the blackboard for working space, and the members write a group solution on the board for each problem. The teacher moves from group to group, checks the proofs (or solutions) on the board, and gives suggestions for improvement. If a group is having trouble with a problem, the teacher might ask a discerning question or, if necessary, give the group a hint.

A group of students will solve a problem that each of the group members would give up on if he were working on it by himself. Each problem is usually designed so that a small group can solve it within one class period. Students in the group usually work at the board on problems they have not seen before the class period. The teacher should assign additional homework problems to be solved by individual students outside of class.

At the beginning of a semester, the teacher should tell students how to cooperate in a small group. Within a group, it is intended that everyone participates and that no one dominates the discussions. There is no appointed leader of a group. A student can ask questions within his group without risking embarrassment in front of the teacher and the whole class; the other group members try to provide satisfactory answers. All the group members must understand a proof or problem solution before the group attacks another problem.

A group with five or more members is too big for solving mathematics problems. In such a group, one or two people tend to dominate, and the rest tend to feel reluctant to participate and ask questions.

The members of a group develop their own intra-group interactions. The teacher has at best moderate control of what happens in a group. The teacher can have an impact by judiciously changing some group members and by giving suggestions on how to improve the interaction in a group.

The issue of how best to divide a class into groups is largely unexplored. The teacher has basically three options: (1) He can choose the groups himself according to his own biases; (2) he can let the students choose their own groups; (3) he can insist on large scale group changes at the beginning of the semester, so that each student has a chance to work with many others. The students can then choose their own groups.

A group which is proceeding at a considerably slower pace than the other groups in the class probably should be split up, and its members distributed in a suitable manner among the other groups.

A teacher can work most comfortably with three or four small groups. He can handle six groups, rather than four, but it is not as much fun and the groups are more on their own. We note that with more than six groups, the need increases for an assistant who can work with the groups.

The small group discovery method tends to develop interest and skill in cooperating with other people to perform a task. It appears that this method also develops individual ability and confidence in solving problems. Moreover, most students enjoy learning mathematics by the small group method.

The small group discovery method has wide applicability with students of varying ability levels in undergraduate mathematics courses. This method has been used in a number of courses at the freshman through junior levels, including abstract algebra and advanced calculus. If one wishes to try the small group discovery method in more advanced courses, the students will probably need to meet in small groups for several hours per week outside of class. We have not yet tried this approach, and the suitability of this method for graduate courses has yet to be determined.

**CONTENT CONSIDERATIONS**

In this section we shall discuss various issues related to the organization of subject matter.

The content of a mathematics course is usually organized around:

(i) A syllabus;

(ii) An extensive outline - i.e., an ordered list of definitions, axioms, theorems, conjectures, questions, problems, examples, and counterexamples;

(iii) The filling in of details in the outline - i.e., proving the theorems, solving the problems and conjectures, and checking or constructing the examples or counterexamples.

Under the methods described here, the teacher is responsible for part (I) and part (ii). The students are responsible for part (iii), proving the theorems and working out the examples.

The extensive outline may ask the student to make or check a conjecture or to complete a partially stated proposition. The student must then resolve the conjecture or complete the proposition in a reasonable way, and prove that his answer is valid.

If the students are going to develop the mathematics in a course, including establishing the hard theorems, the teacher must carefully construct an extensive outline and a syllabus which will motivate the material and help the students develop their intuition about it. This is more easily said than done; it is harder to do in some courses (e.g., real analysis) than in other courses (e.g., point-set topology).

We heartily endorse the statement below by E. H. Moore [5]. (E.H. Moore was one of the teachers of R.L. Moore at the University of Chicago.)

"The teacher should lead up to an important theorem gradually in such a way that the precise meaning of the statement in question, and further, the practical - i.e., computational or graphical or experimental - truth of the theorem is fully appreciated; and furthermore, the importance of the theorem is understood, and indeed the desire for the formal proof of the proposition is awakened, before the formal proof itself is developed. Indeed, in most cases, much of the proof should be secured by the research work of the students themselves."

Constructing a well motivated extensive outline is a difficult task involving many mathematical and pedagogical judgments. An instructor learns how to do this by seeing examples of what other teachers have done and by experimenting with his own classes. Some examples are included in the appendix of this paper. An excellent discussion on motivating mathematics can be found in Polya's books [6].

This discussion should make it clear that one does not write an extensive outline by simply borrowing the list of definitions and theorems from a standard textbook. Thus, an instructor using these methods has considerably less freedom in choosing an extensive outline and a syllabus than an instructor who gives lectures.

We have a few minor suggestions about motivating material and helping the students develop an intuitive grasp of it:

- The instructor should not ask students to prove a theorem about "an empty set". This means: avoid a theorem if the students know very few examples in which it applies and already know that the theorem is valid for these examples. For instance, do not teach the Sylow theorems if the only finite groups that the students are familiar with are Z
_{n}. First explore many other finite groups; then teach the Sylow theorems when there is a use for them. Similarly, do not make a big fuss about embedding an arbitrary integral domain in a field if the only pertinent integral domains which students have seen are the integers and polynomials. - Do not ask students to develop the integers as equivalence classes of ordered pairs of natural numbers. In general, don't assign a theorem which forces the student to wipe out what he already knows, to pretend he doesn't know it, and then to prove the "obvious" in a difficult fashion.
- Teach general topology only after students are familiar with the advanced calculus topology of the plane (including its use in advanced calculus). Speaking globally, rather than locally, abstraction should follow concrete experience.
- When studying E
^{n}as a vector space, the vectors may be represented as*n*-tuples and as lines with arrowheads. All the calculations and proofs can be done with*n*-tuples. But, in order to provide geometric intuition, it is essential to include the description of vectors as lines with arrowheads. (Dealing with lines and arrows for a full month in a mechanics course is a big help to the student.)

The remainder of this section pertains to the organization and to the level of difficulty of the extensive outline.

The teacher should not expect students to prove theorems which are above their level of sophistication. In particular, one cannot expect students to prove many theorems in a course in which they are not usually required to prove theorems on tests, e.g., freshman calculus. The class time spent on what does and does not constitute a proof will greatly reduce the time available for learning calculus computations and problem solving techniques.

Therefore, freshmen should be mainly asked to solve problems. Sophomore should be asked to prove a moderate number of unsophisticated theorems. Juniors in advanced calculus and abstract algebra should be initiated by being required to produce proofs on a large scale.

If the students are to prove the theorems in a course, it is essential that the teacher break the difficult theorems into lemmas which are bite-size pieces. Examples are given in the appendix.

Before assigning a theorem, the teacher needs to have a clear understanding of the proof, including all the details. If the pieces or lemmas for a theorem do not quite fit together, the students will spend much time getting lost and becoming frustrated. Moreover, if a statement or part of a proof is "obvious" to the teacher, but he does not know how to prove it easily, then the students are not likely to find an easy solution either. Thus, considerable difficulties will arise if the extensive outline does not fit together properly.

Here are some further suggestions for writing an extensive outline:

- Do not ask the students to investigate the boring or complicated details that are usually omitted or presented superficially in a lecture class. Be wary of expecting students to work hard on topics they normally sleep through in lecture classes.
- Avoid the standard mistake of "programmed learning": don't give lots of baby-step problems. Lemmas are one thing, trivialities are another.
- Avoid excessive student frustration by assigning many rather simple but not trivial problems, especially in elementary courses.
- Keep the brighter students interested by providing a good sprinkling of challenging problems. The percentage of challenging problems should increase with the course level.

When unforeseen difficulties arise in a problem, an instructor can make in-class changes in the extensive outline by assigning additional lemmas or exercises or by a discussion of the problem. Discussion may include explaining what the problem is all about.

Beginning students often need much help or explicit explanations to get past the "logic" part and onto the mathematical part of many problems. The teacher may have to explain in detail various logical matters, including the mysteries of quantifiers, negation of statements, and proof by contradiction.

In summary, a course taught by these methods is no better than the extensive outline. A poorly constructed extensive outline guarantees a very unsuccessful course. Thus, there is more pressure on the teacher to prepare a carefully detailed and well motivated outline with these methods than in the lecture situation. We speculate that the student's notebook is likely to contain a better outline and poorer explanations with these methods than when the teacher is lecturing.

**PROCEDURES**

In this section we will describe some of the events that occur in the classroom when the instructor is using either the Texas method or the small group discovery method.

In each of these methods the teacher determines the framework of the class by his choice of an extensive outline, by his procedure for introducing new material, and by his style of interacting with the student(s) at the blackboard. The students operate within this framework.

At the beginning of the class period, the teacher may choose to conduct a brief discussion with the entire class. In this discussion he may introduce new material, assign problems, raise questions, and clarify and organize what the students have already done. Some or all of this can also be done in written form.

The library is considered to be "out of bounds" in courses taught by these methods. However, some teachers permit students to read proofs in a textbook *after* the proofs have been presented in class.

During class the students have opportunities to act as teachers and thereby develop skill in presenting proofs in a coherent manner. The students receive constructive criticism from the instructor on such matters as style and comprehensibility, as well as mathematical accuracy. After a student solution is presented, the teacher may summarize its basic points; he does *not* present a more elegant proof himself.

Whenever necessary, the instructor points out errors in student work; he may do this by saying that he doesn't understand how one step of a proof follows from the preceding step. In this manner, the instructor is asking for further explanations when the proof is incomplete or wrong. He also encourages the other students to point out errors or request explanations. Thus, these methods require complete and correct work from the students.

Although the teacher determines the framework for the course, the level of student efficiency determines how much material is covered. A student or group solution that is long-winded, confusing, unnecessarily complicated, or wrong will take up more class time than a "nice" solution. In addition to the presentation time, the total number of theorems correctly proved by the students has an effect on the pace of the course. The pace with these methods is usually somewhat slower than in a lecture course covering comparable material.

At the beginning of a semester the "proofs" presented by inexperienced students will tend to be atrocious. This is to be expected, owing to lack of previous experience in proving theorems and presenting mathematical results. In addition, many students are not used to solving problems they have not already been programmed to solve. Some slow students will start to solve new types of problems only after they have spent several months being confronted with such problems and watching their peers solve them. These are major difficulties of teaching isolated courses by these methods. These difficulties are more severe with the Texas method than with the small group method. Nevertheless, by the end of one semester there will be an improvement in the quantity and quality of student solutions. By the end of a second semester there should be marked improvement.

It is essential that teachers using either method show substantial respect for their students. The interaction between teacher and students is considerably more frequent and direct in these approaches than in a lecture approach, and hence the teacher's style can be more harmful the students' egos. If the teacher acts in an overbearing manner, insults the students, puts them down, or makes mild snide remarks, the students may stop producing mathematics and perform withdrawal actions. Students in the Texas system may try to avoid going to the blackboard. Students in the small groups may work more slowly and sometimes erase a problem solution before the teacher can check it.

It is essential that the instructor has confidence that his students can learn to solve hard problems; he must also have the patience to wait for them to develop this ability. He must have the patience not to explode when a student does something foolish, and often the patience to wait for students to solve a particular problem. Thus, an instructor who has only average patience with students should continue to use the lecture method.

**CONCLUDING REMARKS**

It is not a simple matter to begin teaching courses by these methods. It greatly helps to talk with experienced practitioners, to observe classes in action, to examine some well-constructed extensive outlines, and to read the little material that is available. Prior experience as a student in either method is possibly the best preparation. A teacher operating without these benefits has a distinct disadvantage during his first attempts.

A teacher should use these methods for courses in which he is already an expert. If a motivation for teaching a course is to review the material himself, then he should not use these methods. An exception can only be made in undergraduate courses when the teacher has a reasonable grasp of the course content and then begs, borrows, or steals a first class extensive outline from somebody else and works through all the problems himself, carefully and completely.

As a consequence of the large amount of student-teacher interaction, the instructor will find that these methods of teaching are both more enjoyable and more tiring than lecturing. The lecturer's enjoyment of being an orator or actor is replaced by the enjoyment of interacting with student, seeing them do good work, and learning the new proofs invented by students. Moreover, the teacher in these methods receives immediate feedback on student learning. He can gauge the students' progress and sense of involvement. Thus, the teacher cannot "snow" students for long without knowing it.

In closing, we note that these methods accomplish some important purposes of mathematics instruction. These methods can increase student ability in solving problems and proving theorems, in making conjectures, and in presenting a coherent argument. In our opinion, these advantages more than compensate for the fact that students are usually exposed to somewhat less material than in a lecture course. We modestly suggest that each mathematics major should have the opportunity to take at least one course taught by each of these methods during his undergraduate career.

**REFERENCES**

1. Neil Davidson, *The Small Group-Discovery Method of Mathematics Introduction as Applied in Calculus* (doctoral dissertation, University of Wisconsin, Madison, 1970), Technical Report of the Wisconsin Research and Development Center for Cognitive Learning, Madison, 1971.

2. ------, The Small Group-Discovery Method as Applied in Calculus Instruction, *American Math Monthly* 78 (1971) 789-791.

3. Douglas Robert Forbes, The Texas System: R.L. Moore's Original Edition, doctoral dissertation, University of Wisconsin, 1971.

4. Edwin E. Moise, Activity and Motivation in Mathematics, *American Math. Monthly* 72 (1966) 407-412.

5. E. H. Moore, On the Foundations of Mathematics, *American Math. Soc. Bull.* 9 (1903) 402-424.

6. George Polya, *Mathematical Discovery*, two volumes, New York: John Wiley & Sons, Inc., 1962.

7. Lucille S. Whyburn, Student Oriented Teaching -- The Moore Method, *American Math. Monthly* 77 (1970) 351-359.

8. R.L. Wilder, Axiomatics and the Development of Creative Talent, Amsterdam: Talk published in *The Axiomatic Method*, Edited by G.L. Henking, P. Suppes, and A. Tarski, (1959) 474-488.