Prof. Gangaram S. Ladde
Dept. of Mathematics and Statistics
University of South Florida.

Problem Solving Process


The goal of this work is how to develop and how to foster the problem solving process. Observing the problem solving process as a puzzle solving/game playing process, we make a serious efforts to provide a detailed conceptual procedure of the development, understanding and applying the proposed problem solving process in a systematic and unified way. In this context, the concept of decomposition-aggregation problem solving process is introduced, and its strategy is outlined.

G. S. Ladde
Department of Mathematics and Statistics
University of South Florida
4202 East Fowler Avenue, CMC 342
Tampa, Florida 33620-5700 USA

ABSTRACT: The goal of this work is how to develop and how to foster the problem solving process. By observing the problem solving process as a puzzle solving/game playing process, we make serious efforts to provide a detailed procedure for the development, understanding and applying the proposed problem solving process in a systematic and unified way. In this context, the concept of decomposition-aggregation problem solving process is introduced. Furthermore the decomposition-aggregation problem solving strategy has been outlined in a systematic way.

1.INTRODUCTION: A junior/senior level undergraduate course, MA-460-Problem Seminar with Applications course in the Department of Mathematics, the State University of New York at Potsdam in 1975 [5] was developed. The author is grateful to Professor Clarence F. Stepenens and his colleagues, eventually, this proposed new course became a regular course in the department. In fact, this course further strengthened the newly developed “Four Year Bachelor’s-Masters Degree Program [7]” that was initiated by Professor Clarence F. Stephens in the Department of Mathematics, SUNY-Potsdam. The success of program was a topic of discussion during 1980’s and 1990’s at the National level [5]. The course was composed of the following basic components: (i) training for a mathematical problem solving process, (ii) promoting understanding of a mechanism in the problem solving process, (iii) fostering student interests in mathematics, engineering and sciences, (iv) developing mathematical model building and critical thinking abilities, and (v) instilling self-esteem and confidence-“CAN DO”. The problem seminar course was divided into two parts, namely, (1) General Problem Seminar and (2) Special Problem Seminar. The general problem seminar consists of solving a set of problems that are based on the basic course work in mathematics (Calculus, Linear Algebra, Introduction to Abstract Algebra and Advanced Calculus. On the other hand, the special problem seminar part consists of (a) an individual study of a special topic of student interest in mathematics and (b) a topic of interest in any one of the fields in biological, business, chemical, economic, educational, engineering, medical, physical, and social sciences. At the beginning of the semester, each student needed to provide her/his two personal topics interest: one in mathematics and another other than mathematics. Students’ personal interests and desires are important factors for the success and fun/enjoyment in the course. Within the first two weeks, each student makes her/his own choice of topics (one in mathematics and other in other areas of subject matter). Otherwise, based on the students’ interests, instructor (author) recommends the topics for their choice.

2.LECTURES: The instructor (author) gave 9-10 hourly lectures in the class. The
lectures cover the outlines of the following topics [1,3] with detailed illustrations:

1. Problem Solving Processes

2. Description and Mechanism of Problem Solving Processes

3. Process of Developing Maturity (at the student level background)

4. Collecting and Organizing Material

5. Style of Research Project

6. Presentation of Report

About 6-7 classroom contact hours are used to go over the solutions of general problem sets in the class. Most of the time, students presented their solutions under the supervision of the instructor (author). Students are encouraged to take active participating role in the discussions. During the last 6-7 weeks, students begin presenting their special topic research findings. In addition to open unlimited office hours, the remaining contact hours used for “Mathematical Clinic” that is developed by the author in 1975 [5].

3. MATHEMATICAL CLINIC [5]: The mathematical clinic is a kind of mathematical instructions featuring the problem reading and understanding, identifying the symptoms/difficulties, the critical problem analysis, discussion of a specific problem statements with given pieces of information and goal(s), and solution strategies. Furthermore, it provides facility for mathematical diagnosis and treatment for a student who has difficulty in solving he problem. All students are strongly advised to participate in the mathematical clinic, even though they are not sick of problem(s). The procedure for operating mathematical clinic is as follows: A student is expected to explain (if possible) her/his difficulty in a problem solving process; otherwise, an instructor (author) begins asking question or a series of questions to identify and to understand the level and kind of difficulty in solving the problem. After identifying the level of difficulty, the student will be given a hint to solve that particular level difficulty. He or she is expected to spend some time working on it, and if he/she still has some difficulty, then the instructor will provide another hint and the student will again recommend to work on it. If he/she is not able to complete the solution, then the instructor demonstrates how to use the given hints that were provided earlier to solve the difficulties in the problem solving process.

Two of the most byproducts of the “Introduced Mathematical Clinic” are the processes of problem solving mechanism and problem solving strategies [1,3]. We note that the presented clinic is designed for the conceptual understanding and its applications rather than mechanical drilling and memorizations. Through the individualized efforts and instructions, the role of the instructor is to provide training towards the problem solving process. Moreover, the mathematical clinical approach fulfills at least three most important objectives [1,3,5], namely, (i) to eliminate the fear and anxiety in mathematics, (ii) to develop the model building abilities in the context of the student’s personal interest and hobbies, and (iii) to instill self-esteem and self-confidence-“CAN DO”.

4.RESEARCH PROJECT: The instructor plays a role of gardener nurturing each student at every stage of her/his “research activity”. At every stage, each student is advised to keep notes of read and developed material in both selected topics of their individual choice. We recall that students’ research work begins with the identification of their own personal enjoyments, hobbies, interests and/or skills other than mathematics. Based on students’ choice, each student begins reading the material in the literature (if any) and interacts with the instructor, frequently (with open office hours). Moreover, each student is given a few tips how to read and how to make notes about the reading material. Students are suggested to read certain relevant material in the authors lecture notes [4], and also search the additional related material on their topics of interests. The reading of newly searched material will be supervised by the instructor to make sure that the work is progressing in satisfactory manner with fun and enjoyment in the selflearning process. The students are constantly encouraged to update readings to the instructor. Depending on the student’s internal drive and the ability of handling challenges that are based on the established Mathematical Clinic in Section 3, each student is encouraged to raise her/hi curiosity about developing a mathematical model of a process (dynamic/static) of her/his selected topic of choice. Under the supervision of the instructor, the validity of the mathematical model is examined. Depending on the extent of the originality of the model (at the student’s background level and abilities), each student encouraged to seek a mathematical solution/representation of the process under her/his study. Each student is challenged to critically analyze the process of model building and solution procedures. The project is guided under the presented “Mathematical Clinic of Section 3” approach initiated by the author.

5.ANTICIPATED OUTCOMES: In this course [5], students are expected to exhibit the following abilities with respect to the general problem solving process [1,3,5]:

1. Mathematical Reading and Writing Capabilities

2. Understanding Statements of the Problems in the Sets

3. Knowledge and Understanding of the Basic Ideas Encountered in their Course Work

4. Clarity and Completeness of Thoughts

5. Interest in Problem Solving Process

6. Development of Problem Solving Strategies

In particular, under the supervision of the instructor, each student is expected to complete the following tasks regarding the independent study project [5]:

1. to collect the material regarding their own selected independent study topics the special research topics

2. to read and to understand the material, thoroughly

3. to formulate and to attempt to solve special problem (s) based on the study of the special topics

4. to write a report according to the format provided by the instructor

5. to present the report in the class during the last two week period of the semester

6. to submit two typed copies of the report to the instructor as one of the courserequirements

6.CONCLUSIONS: The briefly presented problem solving process is not only applicable to mathematical sciences, but it is also applicable to problem solving process in any level and any discipline, for instance, biological, chemical, business, economic, medical, physical and social sciences. Moreover, it provides a tool for handling problem(s) that arises at any level such as: home, neighborhood, town, county, state, nation and so on. The continuous supervision and assistance are needed to nurture, monitor and foster the initiated gains made by using the problem solving approach: (a) critically analyzing, developing, formulating, reading, reviewing, and writing the material, and constructing problem strategies, (b) collecting and organizing the material, and (c) utilizing developed problem solving strategies. The problem solving process provides a natural scheme for fostering and strengthening one’s (1) style of solving problem(s), (2) building self-confidence, self-esteem, and self-security, and (3) independence. It initiates, fosters, and strengthens individual’s natural abilities and freedom of choice, actions and reactions to the problem(s) in hand. We further note that the questioning process generates curiosity and creativity, and hence it stimulates an interest in learning and in eradicating one’s ignorance about the subject matter under considerations. Moreover, it naturally motivates to undertake an advance degree education.

7. ACKNOWLEDGEMENT: This work is supported by the Mathematical Sciences Division, U. S. Army Research Office, Grant No.: W911NF-15-1-0182.


1. An Introduction to Differential Equations: Deterministic Modeling, Methods and Analysis (with Anil G. Ladde),Volume-1, World Scientific Publishing Company, Hackensack, New Jersey, 2012

2. An Introduction to Differential Equations: Stochastic Modeling, Methods and Analysis (with Anil G. Ladde) Volume-2, World Scientific Publishing Company, Hackensack, New Jersey, 2013.

3. G. S. Ladde, Problem Solving Process, Bull. Marathwada Mathematical Society, Vol. 2 (2001), pp. 90-104.

4. G. S. Ladde, An Introduction to Biomathematics I, State University of New York at Potsdam, New York, Lecture Notes, 1974.

5.. G. S. Ladde, MA-460-Problem Seminar, SUNY-Potsdam, New York, 1975.

6. John A. Poland, A Modern Fairy Tale, The American Mathematical Monthly, Vol. 94, No. 3 (1987), pp. 292-295.

7. C. F. Stephens, Four Year Bachelor’s-Masters’ Degree Program in Mathematics, SUNY at Potsdam, New York, 1969.