The purpose of teaching mathematics is not only to drill students in routine computational procedures, but also to help them master problem solving. The kinds of problems encountered at the work place are not those found in most traditional textbooks. Thus most of our students are retrained by employers before they can work efficiently. To arrest this problem, the NCTM Standards listed problem solving as one of the four areas that should receive increased attention at all grade levels: K-12. Although students should continue to learn basic computational skills, the Standards recommend that emphasis be placed on problem-solving skills to enable students to read, write, and think mathematically.

The Problem Solving and Mathematical Reasoning Workshop (The Workshop) addressed the National Council of Teachers of Mathematics (NCTM) Standards on problem solving and mathematical reasoning. Twenty elementary and middle school teachers from different schools, representing both public and private schools, were selected to attend a two-week workshop at the University of Tennessee at Chattanooga on June 12-23, 1995. During the workshop, participants were provided with skills, techniques, and materials that they needed to have in order to implement the NCTM Standards on Mathematics as problem solving and reasoning in their classrooms.

In the workshop, we trained teachers to be problem solvers and demonstrated how they should transfer those skills to their students. Specific areas include:

The first day we introduced the teachers to how the brain processes information. The brain's storage house is divided into two: the short-term memory and the long-term memory. When information gets into the brain, it is first stored in the short-term memory. Under conducive conditions, the information is transferred to the long-term memory where it becomes knowledge. The duty of the teacher is to facilitate the transfer of information from the short-term memory to the long-term memory. The teacher must know the appropriate activities and actions that allow the transfer to take place. These activities include well-designed problems that involved writing and thinking, proper questioning and proper attitude towards the students by the teacher. These characteristics are integral parts of the problem solving process.

The first step in solving a problem is to understand the problem. To show an understanding of any problem, the reader should be able to answer the following questions.

c) What is it the reader is looking for, or trying to find?

d) What are the unknowns?

e) What information is given in the problem?

f) Is the given information sufficient to solve the problem?

g) Any redundant, contradictory or missing information in the problem?

The reader may need to read the problem many times to achieve understanding. We used many examples to demonstrate this process to the participants.

The next step, after achieving understanding, is to find a strategy to solve the problem. The teacher should coach the student on how to conceive a plan but not give it to the student. The following are some of the strategies discussed at the workshop:

a) Guess and check. b) Work backwards.

c) Draw a picture. d) Draw a diagram.

e) Use direct reasoning. f) Use indirect reasoning.

g) Find a pattern. h) Use an equation.

i) Make an organized list. j) Solve a simpler problem.

Carrying out the plan involves solving the problem using some chosen strategy. In performing all necessary computations, the student should make sure that each step of the plan is correct as well as keeping an accurate record of the work. If the chosen strategy does not solve the problem, the student should select a new strategy and start all over again. The student must not be ashamed to seek hints from others who can solve the problem. (They should only guide but not solve the problem for the student.) Moreover, it is advisable to give oneself enough time to solve the problem as rushing could lead to unnecessary mistakes.

This step involves

a) Examining the result and interpreting it in terms of the original problem.

Finding out if the answer obtained makes sense or is reasonable.

b) Checking to see if the result satisfies the conditions of the original problem.

d) Finding out if there is another and simpler method of solving the problem.

e) Determining if the strategy can be used to solve other problems.

f) Exploring the possibility of extending the method to solve a more general

problem.

Writing is very important for understanding and for communication. Students should write out the answer to the problem in complete sentences, and specify all units of measurement.

The Standards recommend the appropriate use of calculators at the K-8 grade levels. Participants were taught the proper use of calculators in problem solving. The transfer of such skills to their students was emphasized and demonstrated.

The session on calculators started with a NCTM video showing classroom children using calculators for problem solving and for exploration. On the video were comments by teachers and students on how calculators aid in teaching and learning. This approach was very effective in dismissing bad misconceptions about the use of calculators in the classroom and making participants more open minded.

During the workshop, participants were introduced to different keys of the TI-12 (Math Explorer) calculator. The use of these keys in different problem settings was demonstrated. Each participant's school was given a TI-12 calculator and an overhead. The calculators used in the training were loaned from Texas Instrument's Loan Program.

Many women and minority students lack the skills needed for advanced work in mathematics. The difficulties that women and minority students may have in the study of mathematics, and how to eliminate them, were discussed during the workshop. Ms. Barbara Thomas, then the Mathematics Supervisor for Chattanooga City schools, exposed participants to activities that can be employed in a multicultural setting, activities that get every student involved in the problem solving process. Ms Ava Warren and Ingrid Hutchinson, also gave presentations on women and minority issues. These presentations, apart from helping participants become good role models, increased their sensitivity to the difficulties that students from under represented groups may face in the study of mathematics as problem solving.

The following were our expectations of teachers selected to attend this workshop.

1. Teachers who become more excited about mathematics as problem solving.

2. Teachers who are ready to go back and implement new ideas.

3. Teachers who become leaders in implementation of the NCTM Standards in the

classroom.

4. Teachers who become leaders and present workshops and seminars on the NCTM

Standards on problem solving to others.

5. Teachers who are constantly seeking for new knowledge by attending workshops,

and etc.

These expectations were factors in the selection process. We were delighted that participants met these expectations in a very short period of time.

Each teacher has conducted an in service training to colleagues who are involved in mathematics instruction. Reports form principals show that these teachers are very excited about using the problem solving in their classes.

The importance of involvement in professional development activities was emphasized during the workshop. After the workshop was completed in June 23, 1995, we encouraged many participants to attend other workshops that were conducted by the Chattanooga City Schools. On Oct. 5-7, 1995, some participants attended the NCTM conference in Knoxville on sessions dealing with problem solving. From the grant money, we were able to register 12 participants in the Chattanooga Area Mathematics Teachers Association (CAMTA). Apart from being active members of CAMTA, some of them are actively conducting workshops for teachers in sister schools.

In recruiting participants, our goal was to include teachers from all ethnic backgrounds, the Chattanooga City Schools, the Hamilton County Schools, and private schools in the Chattanooga area. The demography information on the participants is given below. This information indicates that our recruitment goal was achieved.

Minority teachers 45% (9 teachers)

Private school teachers 15% (3 teachers)

Chattanooga City Schools teachers 55% (11 teachers)

Hamilton County School teachers 30% (6 teachers)

The participants and instructors met as a group after the workshop on September 30, 1995. During this time each participant was given an opportunity to share her experience and activities with the rest of the group. Some of the activities shared include using Venn diagram in grade two math, creating mathematics Centers, using games in problem solving, and asking students to write math journals. It is not possible to include in details the experiences of each of the twenty participants. However, we feel that the following should be mentioned.

1. A participant confessed that before she was not a 'math person.' But now she has created a mathematics center for her students. Moreover, she has now discovered a math problem that was at the end of book that is very good but which she did not notice before (despite the fact that she has taught for 4 years using the same book). Another participant also indicated using a problem in her class book that she did not before give any importance to.

2. A participant discussed how she designed a concession stand math problem. The project was to give the class a hands-on experience with figuring out and giving out change.

3. Another participant reported having a better semester than she had in the previous 5 years.

4. Another report was from a participant who said that before her students had to do problems her own way. But now she is more flexible. She doesn't mind any method provided the student explains why.

5. One of the teachers complained that it took so long for her students to complete the five stages of the problem solving process taught during the workshop. She wanted to know what others do to overcome that problem. Many suggestions were given. One of them was that she could just ask the students to only explain why they perform each operation and to write out the solution in a complete sentence. Alternatively, she could emphasize the different steps at different times so as to expose the students to all the problem-solving process.

The workshop brought teachers from different school systems in the Chattanooga area together. The first day of the workshop we asked for and obtained permission to make available to each participant the phone numbers of all participants. We stressed to participants the need to know and be of help to each other. They should be free to call each other or the instructors any time for help. These have been going on very well even with teachers in different school systems. Many are now good friends.

Another positive result of assembling these teachers from different school systems was the opportunity they had to share their experiences. These teachers have students with different backgrounds. Getting to share in the experiences of the others was very significant, particularly as the Chattanooga City Schools and the Hamilton County Schools have agreed to merge in a couple of years.

We were able to raise the level of enthusiasm of participants by our actions and words. We strongly believe that an enthusiastic participant is the one likely to go back and implement the problem solving process. During the follow up activities, we clearly saw it happen. A principal, while evaluating one of her teachers, wrote, "Mary's enthusiasm has spread through my faculty". Other math teachers have required an opportunity to see these activities and Mary is "in action" with her students.

We asked participants to share whatever teaching ideas or activities they had with the group. Five participants volunteered to share some of their activities. We always ask participants to give their sincere opinions on pedagogical issues. This participation by participants was to make them feel at home, and most importantly, for them to know that it is OK for them to be flexible with their students. Evaluations and follow up comments by participants show that this was a good learning experience for them.

We selected teachers who were teaching K-8 to attend the workshop. However, and because we believe in the teacher having a superior knowledge, our teaching materials were the same for all levels. We discovered, fortunately, during the first few days of the program that the K-3 teachers felt left out. The team quickly made changes in the course materials and exercises in order to reach out to everybody. This situation put much stress on the instructors, as each of us had to over work ourselves for the success of the program. From our experience, we feel that similar projects be done for K-3 and 4-8 grades separately. Many of the participants also suggested this arrangement in their evaluations of the program.

Our major thrust in recruiting was to get minority and private school teachers into the program. This we achieved. However, there was no male participant in the program. This was an oversight on our part. We believe that targeting males as minority and private school teachers were targeted could have drawn male teachers to the workshop.

The project team designed and or selected handouts and problem set on problem solving and reasoning. These handouts, problems, and solutions to problems, a total of 287 pages, were giving out to each participant. We were able to loan calculators from Texas Instrument which were used in calculators' activities. The grant also provided money for calculators and overheads for participating schools.

Attracting private school participation was a major goal in our recruitment efforts. Our activities in this direction included the following:

1. The project director visited many private schools in the area to talk to teachers and principals about the workshop. Principals were encouraged to reward attendees appropriately.

2. A week after talking to private school principals, the project director sent letters encouraging them to nominate their teachers for the workshop.

A total of 8 private school teachers (40% of the total number needed) responded to our call. Unfortunately, our hands were tight and we could only accept 3 of them into the workshop.

It is pertinent to point out that private schools' teachers were very happy that we invited them to participate. On one of the evaluation forms, a private school's principal wrote 'we would love to participate in more of these.' Another wrote, 'Thank you for including one of our teachers in the workshop.' These sample comments and personal discussions with private school principals indicate that private school teachers, especially those from not so rich private schools, have been neglected in government professional development programs. Our workshop successfully addressed the lack of access problems, faced by some private school teachers, in the Chattanooga area.

The main purpose of this project was to teach the NCTM standards on mathematics as problem solving and reasoning to elementary and middle school teachers in the Chattanooga area, and to assist them in implementing these Standards in their classrooms. The success of this workshop was evaluated in the following ways.

1. Participants were given an opportunity to evaluate the program at the end of the first and second weeks, respectively. These evaluations were uniformly positive.

2. Participants kept journals in which they made entries weekly. These entries documented where in the curriculum the teachers used the knowledge and methods obtained through the workshop and their utility. These journals were presented at a shared session on September 30, 1995. Each journal presented told a story of hard work, enthusiasm, and success.

3. The director contacted participants and their principals by phone and/ or by site visits to encourage and evaluate implementation of the techniques learned in the workshop.

4. We required that their principals nominate participants, and that principals be involved in the evaluation process. Principals' evaluations were overwhelmingly positive.

5. The participants, their colleagues, and some principals have expressed interest in participating in future project. That some principals wanted their teachers to be included in future projects by itself gives credence to the success story of the workshop.

6. Many participants have made efforts to attend more workshops, get involved in professional development activities, and buy calculators for their classes. (On September 30, one of the participants told the group how she got two parents of her students to buy calculators for the class.)

7. Participants have given in service training to colleagues who are involved in mathematics instruction. A total of 251 schoolteachers have attended these in service training.

Supervisors and principals in both the Chattanooga City Schools and the Hamilton County Schools were very helpful and contributed immensely to the success of the workshop. Some of the more specific help are described below.

1. Ms. Barbara Thomas, the then Mathematics Supervisor for the Chattanooga City Schools, assisted in the recruitment of targeted groups. She was very active in project development and planning. Moreover, she gave some presentations during the workshop.

2. Mr. Don Upton, Mathematics Supervisor for Hamilton County Schools, assisted in the selection of participants from Hamilton County School System.

3. The Chattanooga City Schools System and the Hamilton County Schools System allowed us to use their pony express to send letters to participants and principals.

4. The principals assisted in the selection of participants from their schools.

5. The principals gave participants time to conduct second tier training during their in service days.

6. Some principals provided participants from their schools with activity books and calculators.

The workshop addressed the Standards concerning the problem-solving process and mathematical reasoning. Upon completion, participants returned to their classrooms with knowledge, skills, techniques, and materials that aid the implementation of the NCTM Standards on Mathematics as problem solving and reasoning. Participants were taught the proper use of calculators in problem solving. The transfer of such skills to their students was emphasized and demonstrated.

The difficulties that women and minority students may have in the study of mathematics, and how to eliminate them, were discussed during the workshop. These presentations, apart from helping participants become good role models, increased their awareness of the difficulties that women and minority students may have in the study of mathematics as problem solving.

Moreover, by aiding the implementation of the NCTM Standards, the Workshop provides one route to meeting the Fourth National Education Goal that United States students be first in the world in mathematics and science achievement by the year 2000.