The Workshop Method of Teaching: An Example from the Discipline of Mathematics Education
By Mark Spikell and Behrouz Aghevli

There are significant differences in the teaching methodologies employed by educators in various disciplines. This article provides insight into one methodology known as a workshop method of teaching in mathematics education. The method relies heavily on the involvement of the learner in formulating or constructing knowledge through hands-on, small-group and individual explorations, using concrete objects (manipulatives) or technology. While this methodology is ideally suited to the teaching of mathematics topics and mathematics education courses, it is also readily adaptable in the teaching of other disciplinary topics and courses too.

Some advantages of this methodology include:

1. The workshop method enables the learner to explore or master relatively abstract ideas by first encountering them in concrete, physical embodiments, then as pictorial representations, and finally in symbolic (letter, number, sentence) form. The workshop instructor carefully designs the workshop to present the ideas studied in these embodiments, and generally in the sequence noted -- concrete, pictorial and symbolic.

2. The workshop method helps the instructor create an environment in which the learner is more likely to be involved and motivated. The workshop method focuses on participatory, hands-on learning; small-group activity and problem solving; pair and small-group discussions; etc. As a result, because of the "active" rather than "passive" nature of the experience, larger numbers of learners are motivated to participate and learn.

3. The workshop method enables instructors to function as the "guide on the side," rather than as a "sage on the stage."  Those using the workshop method do not focus on telling students information.  Instead, they essentially create learning experiences that guide, direct, and facilitate the acquisition of new knowledge by the learner.

This latter advantage, in particular, is quite powerful.

Guide on the Side

Many engaged in teaching, whether at the university level or pre-university level, are guided by the desire to be less of a "sage on the stage" and more of a "guide on the side."

The "sage on the stage" is the individual who believes that the primary role of educators is essentially to give or tell students some pre-determined knowledge that is deemed important. Dominant instructional methods employed by such individuals include: lecturing; use of audio/visual presentations; guest speakers; assigning extensive readings; etc. Essentially, instructors are knowledge givers and students are knowledge receivers.

The "guide on the side" is the individual who believes that the primary role of educators is to motivate students to learn on their own methods and techniques for finding, discovering and mastering pre-determined knowledge that is deemed important. Dominant instructional methods employed by such individuals include: cooperative and group learning; mentoring; hands-on workshops; etc.

For those interested in being more of a "guide on the side," teaching using a workshop approach is an instructional methodology worthy of consideration.

While the specific workshop example presented here is in the discipline of mathematics education (herein defined to mean the discipline focused on the methods of teaching mathematics at pre-college levels), readers will easily see how elements of the workshop methodology can be applied or adapted to other disciplines. And, to encourage a wider readership of the paper beyond just mathematically and scientifically oriented persons, the mathematics discussed in the article is appropriate for middle school, pre-algebra or beginning algebra students. Thus, readers can be assured that knowledge of mathematics beyond the simplest, basic algebra is not required in order to follow the development presented and to gain insight into the workshop methodology employed.

Conceptual Background

As background, readers not familiar with the discipline of mathematics education in contrast to the discipline of mathematics might like to know in advance of reading the paper that most courses in mathematics differ from most courses in mathematics education in at least two important ways.

First, most mathematics courses in a School or College of Arts and Sciences or Engineering are taught either to majors in the discipline or as service courses or requirements for students from many other disciplines. And, these students have a wide variety of future career goals. In contrast, mathematics education courses are taught in a School or College of Education to a specialized population of students, almost all of whom are or intend to be teachers of mathematics in the elementary grades K-4, the middle school grades 5-8, or the secondary grades, 9-12.

Hence, in mathematics education courses, even those that might be taught within a School or College of Arts and Sciences, there is a clear and definite focus on more than just the knowledge normally associated with the content discipline. There is actually a combination of foci in the usual syllabi including:

1. presentation or review of appropriate requisite mathematics content for teaching mathematics topics at school levels;
2. presentation of appropriate methods for presenting (teaching) or reviewing mathematics content topics regardless of the age of the learners;
3. presentation of a variety of materials (physical manipulatives, e.g.) and resources (technology, e.g.) which facilitate the teaching and learning of mathematics content topics;
4. presentation of strategies for developing, organizing, and managing the teaching and learning of mathematics content topics, including how to develop lesson plans, lesson and unit presentations, etc.

Second, students taking mathematics education courses are preparing for a specific career goal of teaching mathematics to others at the pre-college level. Thus, an important function of those teaching mathematics education courses is to model in their teaching the various methods their students are then expected to employ in their own careers as teachers. And, therefore, because these future teachers are expected to use hands-on activities, involving concrete objects or technology with small groups, the workshop method is particularly suited as a dominant instructional methodology in courses they take. That would not be the case in most mathematics content courses, at least not as the dominant or primary method of instruction. While an occasional workshop might be appropriate, in most mathematics courses other instructional methodologies are the general means of instruction.

The Paint Problem

The specific The Paint Problem exploration provides an example of what it means to be a "guide on the side," creating an environment whereby students actively engage in discovering knowledge and information more or less on their own, but under the guidance or mentoring of the instructor. 

The Paint Problem itself is adapted from Robert Rey's article, "Discovery with Cubes" (1988).  Using a manipulative (in this case a collection of cubes), students are guided through hands-on activities to the discovery and statement of 12x(n-2) as the solution to a question posed in the Paint Problem exploration.

An extension is also presented, for interested readers, to show the discovery and statements 6x(n-2)² and (n-2)³ that can also result from the exploration described.

In this early web version of the Paint Problem exploration, the activities in the workshop are described and readers can "do" them with real cubes, though that is not necessary to follow the development. In a later web version of this exploration, readers will be able to "do" the activities on-line, using a virtual version of the manipulative we are developing here at George Mason University in the Metamedia Mathematics Program's Dr. Super's Virtual Manipulatives Project (http://www.galaxy.gmu.edu/~drsuper).

The Paint Problem workshop has been presented several times in the course, EDCI 609: Problem Solving in Middle School Mathematics, offered in the Graduate School of Education at George Mason University. It has also been presented in a number of professional development courses or learning experiences for Virginia math lead or resource teachers in grades 4-8. Students or teachers in the courses and the instructor literally do in the classroom the sequence of activities described here. Essentially, a handout describing the sequence of activities is given to participants after they have participated in the workshop learning experience. And, it accurately summarizes for them what they have done in the class setting.

Thus, the workshop in which they participate provides the opportunity to discover on their own, but with guidance, pre-determined information. And, the summary of their workshop experiences given to them as a handout following the experience provides one means by which learners can verify, review, analyze, and internalize what has been learned during the workshop.

The Paint Problem Exploration

This handout provides a detailed summary of the Paint Problem exploration presented in the workshop. The summary reviews the following:

1. The sequence of tasks (activities) in the exploration

2. The methods and techniques for introducing the activities in a class

3. The important content topics that might be developed through the activities

4. The National Council Teacher's of Mathematics (NCTM) Curriculum and Evaluation Standards and the Virginia Mathematics Standards of Learning (SOLs) that might be addressed through the activities.

Also included are some of the frequently asked questions, with answers, raised by teachers and other professionals who have explored the Paint Problem in a workshop or tried it with their classes.

Permission is given to educators to use the handout for appropriate non-profit educational purposes.

The Sequence and Suggested Methods and Techniques for Introducing the Paint Problem

1. Place students in groups of 2-6.

2. Give each group 225 cubes. (Refer to Frequently Asked Questions)

3. As a large group activity, ask each student to pick up one cube. In turn, ask each of the following three questions, allowing enough time for students to answer the question and for you to review the pertinent facts prior to the start of the actual activity. (Refer to Frequently Asked Questions)

1. A cube has how many faces?               Ans. 6 faces

2. A cube has how many edges?              Ans. 12 edges

3. A cube has how many vertices?           Ans. 8 vertices

4. Explain that you will refer to the cubes they are holding as unit cubes. They are called unit cubes because, in this activity, the length of each of the 12 edges is assigned a measure of 1 unit.

5. Instruct the groups to build a 4x4x4 cube.

6. Once all (most) groups have constructed the 4x4x4 cube, give the following instructions:

"Imagine that your cube was dropped into a bucket of paint and completely submerged. How many of the unit cubes are painted on only two sides?"

7. Tell students to jot the number on a piece of scrap paper and have a group representative bring the paper to a central "data collection" place (teacher's desk, a table, etc.) once they have determined the number. (Refer to Frequently Asked Questions)

8. Once the assigned task has been completed by a sufficient number of groups, record the data collected on the blackboard or an overhead projector for all to see. See Figure 1 which shows data actually given in a 5th grade class. Notice that the data is not organized. That is, it is written just as it is read from the student papers. There is no attempt to group the data, label the data, or organize the data in any way. At this point in the activity, no further comments about the data should be made. Merely record the data on the board for students to see and move on. (Refer to Frequently Asked Questions)

9. Next, instruct the groups to build a 5x5x5 cube, imagine it was dropped in the bucket of paint and completely submerged, determine how many of the unit cubes are painted on only two sides, record the number on a slip of paper, and bring the paper to the central "data collection" place.

10. Add the data for the 5x5x5 cube to that recorded for the 4x4x4 cube on the blackboard or overhead. As before, record the data in a random and unorganized manner with no labeling, etc. See Figure 2.

11. Next, instruct the groups to build a 3x3x3 cube, imagine it was dropped in the bucket of paint and completely submerged, determine how many of the unit cubes are painted on only two sides, record the number on a slip of paper, and bring the paper to the central "data collection" place.

12. Add the data for the 3x3x3 cube to that already recorded for the 4x4x4 and 5x5x5 cubes to the blackboard or overhead. As before, record the data in a random and unorganized manner with no labeling, etc. See Figure 3.

13. Once the data for the 4x4x4, 5x5x5 and 3x3x3 cubes has been collected, make a table for all to see. For the first time, actually provide some organization and labeling as shown in Table 1. (Refer to Frequently Asked Questions)

14. Next, instruct each student to make a freehand sketch of a 6x6x6 cube. [Alternatively, give students an already drawn sketch of a 6x6x6 cube.] Tell them to use the sketch to help determine how many of the unit cubes would have only two faces painted if the 6x6x6 cube were submerged in paint. [Note: For some students, it will be a good exercise to have them actually shade in or darken the visible faces of the unit cubes that would have only two faces painted. It can be instructive to engage them in a brief discussion of how doing this might be useful in answering the question.] (Refer to Frequently Asked Questions)

15. After a suitable period of time, ask for and add the data to the table as shown in table 2.

16. At this point, invite students to examine (analyze) the data in the table. Ask them, for example, "What patterns and relationships can you see?" Some of the responses you can anticipate include:

          A. In the unit cubes column the numbers differ by 12.

          B. In the size of cube column the numbers increase by one each time.

          C. In the unit cubes column the numbers are all multiples of 12.

          D. In the unit cubes column each number is the previous number plus 12.

17. If for any reason either observation in C or D is not forthcoming, guide students in some appropriate way to recognize these patterns.

18. It is important at this point to record symbolically some of the patterns and relationships observed by students. See Table 3.

* Thought process A represents the thinking of those students who saw that each entry was some multiple of 12.

** Thought process B represents the thinking of those students who saw that each subsequent entry was the previous entry plus 12.

19. Next, pose the question, "Based on the information in the table, can you predict the number of unit cubes with only two faces painted for a 7x7x7 cube?"

20. Invite student answers for the 7x7x7 cube. Ask students to give a reason for their prediction. There should be several different explanations as students will not all solve the problem by thinking in the same way. Make an effort to elicit from the students as many explanations as time permits. If any ways of thinking about the problem arise that are different from those shown in Table 3, add them to the table for all students to see. Such an additional thought process is shown as Thought Process C in Table 4.

* Thought process C represents the thinking of students who saw that each entry was 12 + some multiple of 12.

21. Tell students to make and use a sketch of a 7x7x7 cube (or an already drawn version you can distribute). Tell them to shade the visible faces of the unit cubes that would have only two faces painted. Have them use that information to verify their prediction.

22. From this point on, do as little or as much additional data collection as necessary for the particular class of students. By this time, many students will be able to recognize, write and/or verbalize additional relationships and patterns, most notably the generalization that for an n x n x n cube, the number of cubes with only two faces painted is given by 12 x (n - 2). Generating this linear relationship from an activity with cubes fits nicely into one of the major topics in any pre-algebra or algebra class, Functions.

23. An excellent final question to pose for this activity is, "Can anyone provide an explanation or 'proof' that 12 x (n - 2) gives the number of unit cubes that will have only two faces painted?"  In many groups, one or more students will be able to give a fairly sophisticated explanation along the following lines. For any n x n x n cube, the unit cubes along an edge (except for the two on the ends) have only two faces painted. Thus, on a given edge, there will be n - 2 such cubes. Since there are 12 edges in a cube, 12 x (n - 2) must give the total number of unit cubes that will be painted on only two faces.

24. The natural extension of this activity asks the questions, "How many of the smaller cubes would have 3 faces painted? ... , only 1 face painted? ... 0 faces painted?" And, whether to ask all of the questions at once or only one at a time in sequence depends upon the teacher's goals for the lesson. For example, if the goal is solely to discuss linear relationships, then stopping at this point is appropriate. However, if the goal is to show that there are other relationships (e.g., quadratic, constant, or cubic) then posing one or more of the additional questions makes sense.

Extending the Paint Problem

Each of the extensions could be developed individually or collectively in much the same way as that outlined above. Table 5 summarizes the data that would be collected in determining how many of the smaller cubes have 3, 2, 1 and 0 faces painted.

Table 6 shows some of the patterns that students should recognize when they examine the data in the table.

Table 7 gives the conclusions that students should be able to make from the patterns in Table 5.

It is within the grasp of some students to actually develop the explanation or proof that (n-2)³ and 6x(n-2)² are correct.

Standards and Learning Goals

The Paint Problem can be used by teachers to motivate specific learning outcomes in mathematics or problem solving in general. For example, several of the ideas that can be motivated or discussed in conjunction with the Paint Problem include:

1. Collecting, organizing and analyzing data

2. Forming and verifying an hypothesis

3. Stating a conclusion

4. Using appropriate notation

5. Developing algebraic thinking

6. Writing a linear expression or equation

7. Recognizing patterns

8. Reviewing factors and multiples

Importantly, the Paint Problem also effectively illustrates each of the four core NCTM and Virginia SOL standards of Problem Solving, Reasoning, Communication and Connections.

Frequently Asked Questions

This Paint Problem has been used many times with students or teachers. Consequently, a number of questions have been asked by others that may be of interest to current readers. Some of these questions and answers follow.

Question: In task 2, why does it call for 225 cubes? 
Answer: During the course of the workshop, students will be constructing five cubes, from a 1x1x1 to a 5x5x5.  Having all five of the cubes standing on the table at the same time has added value. For many students, this may be the only opportunity they ever have to see the exponential growth that occurs from 13 to 53. Since, the five cubes require 1, 8, 27, 64, 125 cubes, respectively to build, they will need a total of 1+8+27+64+125 = 225 cubes.

Question: In task 3, what if students ask (or do not know) what a face, edge or vertex is? 
Answer: A technique I have used effectively is not to define the terms. Instead, I usually just point to the faces (or edges or vertices) on a unit cube I am holding in my hand and say, “These are faces (or edges or vertices).”  Then I then move on rapidly to the next step in the activity. Most students grasp the idea and I do not become bogged down in trying to formalize a definition. In this sense, I really try to put into practice the philosophy that a teacher is a ‘guide on the side, not a sage on the stage.’ So, I try to avoid telling students very much at all, whether their questions are about definitions of terms, instructions to activities, etc. Instead, I try to help students discover or see for themselves what the answers to the questions might be by guiding them to formulate their own understanding. I do this through such actions as pointing, posing additional questions, or eliciting responses from students.

Question: In task 7, what is the purpose of having students bring scraps of paper to a place in the room? Why not just ask for the answers while they are at their seats? 
Answer: I have found it difficult to time various group activities. Even when moving around the room observing the various groups, there are times when I am just not certain which groups have completed a task and which have not. Also, I have found that some groups have no sense of the need to bring a task to conclusion. Left to their own devices, the task would go on and on and on. 

The technique presented gives me a good way to both control the timing of the activity and also to give a means for motivating most groups to complete the task in a timely fashion. As group representatives begin to come forward, it sets a move on to conclusion tone for the other groups. And, at some point, when enough groups have actually come forward (i.e., actually completed the task), I can readily conclude that moving on is in the best interests of all students. By the way, I usually use the 80% - 20% rule as my guide. That is, when about 80% of the groups have finished a task, it is time to move on. The others can catch up by seeing the results of the work of other students. 

Question: In task 8, what is the reason for recording the data in a random, unorganized way with no labeling? 
Answer: I am again attempting to be the guide on the side, not the sage on the stage. That is, to the extent possible, I am trying to minimize the structure and information giving that I do as the teacher. Instead, I am choosing to provide the minimal structure and information necessary for the students to be engaged in the activity and to possibly formulate for themselves the organization necessary to understand the data. Should that fail, I can bring the necessary order into the picture later.

Question: In task 13, because you are only using some of the data given by students, what do you say if students ask, “Is the data in your table correct?” 
Answer: Because there are incorrect pieces of data in the unorganized lists already on the board, some students may indeed ask this type of question. And, in fact, I place the incorrect data on the board in previous steps in the hopes that this question might arise. It gives me an opportunity to share with students in a meaningful context the point that not all data collected in a problem is accurate, needed or necessary. In fact, very often there is extraneous or incorrect information present. We need to be aware of that fact and ever alert to deciding what is necessary, what is extraneous, and what is accurate or inaccurate.

Question: In task 14, what is the value of having students sketch the 6x6x6 cube and shade the visible faces of the unit cubes with only two faces with paint? 
Answer: First, an important goal of school mathematics is the development of spatial sense. One aspect of spatial sense is the ability to visualize three-dimensional shapes from two dimensional sketches, and vice versa. This gives students exposure to that aspect of spatial sense. 

Second, the pictorial representation of the cube provides an important embodiment for the eventual internalization by students of the ideas being explored. That is, I am a firm believer that if you want a student to grasp (internalize with understanding) an abstraction (i.e., a mathematical idea or skill), then it should be presented in concrete, pictorial and symbolic forms as part of the developmental process. If this is done, learners are better able to ultimately construct for themselves a meaning or understanding of the abstraction. Building the cubes is the concrete embodiment. Working with the sketch is the pictorial embodiment. 

Third, the act of shading the faces provides an organizer for some students. This organizer effectively may guide them toward an approach to thinking about how to determine the number of cubes with only two faces painted. That is, some students may recognize or discover through the act of shading that the unit cubes with two faces painted all fall along edges of the larger cube. For many students this is not an obvious fact. And, giving them the opportunity to discover it is superior to telling it to them.

The Paint Problem described in the workshop is adapted from an article, "Discovery with Cubes," by Robert E. Reys (Mathematics Teacher, May 1988, pages 377-378). 

References

Reys, Robert E. (May 1988). "Discovery with Cubes," Mathematics Teacher, pp. 377 - 378

Dr. Mark A. Spikell (mspikell@gmu.edu) is Professor of Education (Mathematics) at George Mason University (GMU) in Fairfax, Virginia. He has an Ed.D. in Curriculum, K - 12 (Mathematics), from Boston University; a B.A. in Mathematics from Miami University; and, 35⁺ years experience as a mathematics educator in schools and universities. He is author of Teaching Mathematics with Manipulatives, (Allyn and Bacon, 1993), and co-author of more than 27 resource books, curriculum units and articles. He has been a frequent speaker at professional meetings and active in many organizations, including the National Council Teachers of Mathematics, the National Council Supervisors of Mathematics, the Association of Mathematics Teacher Educators (past president), the Virginia Council Teachers of Mathematics and the Association of Teachers of Mathematics in New England. Currently, he is Coordinator of Ph.D. and M.Ed. Leadership Programs in Mathematics and Science Education and Director of the Metamedia Mathematics Program at Mason.  In 1995, Dr. Spikell received the George Mason University Teaching Excellence Award.

Dr. Behrouz Aghevli (a.k.a Dr. Super) has a Ph.D. in Mathematics from Northwestern University.  He is a Senior Information Officer at the World Bank and Affiliate Associate Professor of Education (Mathematics) at George Mason University.  Dr. Aghevli has invented several patented manipulatives and has co-authored teacher resource books including: Super Math with Terrific Triangles: Hands-on Mathematics for First Grade (Scott Resources, Inc., 1999, with M. Spikell and C. Talbot); Discovering Greatest Common Factor and Least Common Multiple with Dr. Super’s Factor Blocks (ETA , 1998, with M. Spikell and N. Klimenko); Exploring Trigonometry with Dr. Super’s Trigrams (Stokes Publishing Company, 1998, with M. Spikell and C. Roller);  Dr. Super’s Triangles: Fraction Explorations (Creative Publications, 1995, with M. Spikell);  Triango: A Guide To Strategy Explorations (Creative Publications 1994, with M. Spikell).  He has been a regular speaker at regional and national meetings of the National Council Teachers of Mathematics since 1994.