Skills Needed for Mathematical Problem Solving (Part 7)

This concluding section examines the role of group work in mathematical problem solving, summarizes the paper's conclusions, and provides the complete reference list along with the full solution to Problem 6 (the two-boats river problem). Together, Parts 1–7 present a comprehensive research framework for understanding the skills, attitudes, and practices that support genuine mathematical problem solving.

Table of Contents

3.6 Group Work and Problem Solving

Experience shows that group work is very useful in solving problems in general. When the problem given to students is challenging, students are happy to work in groups. In fact, it was shown that cooperative learning and metacognitive activities have positive effects on the students' abilities to solve problems. Group work also prepares students for the future where they have to work together on large problems and projects.

Benefits of Group Work in Mathematics

  • Diversity of approaches: Students see multiple strategies for the same problem.
  • Metacognitive development: Explaining reasoning to peers forces students to become aware of their own thinking.
  • Motivation: Challenging problems feel less intimidating when tackled collaboratively.
  • Communication skills: Students learn to articulate mathematical ideas clearly.
  • Professional preparation: Mirrors real-world collaborative work on large projects.

Conclusions

Many skills and factors are involved when genuine mathematical problems are being solved. Instructors have to understand and be familiar with these factors and skills. They also need to design activities and guide students to develop and use these skills. It is also possible to design problems that focus on a limited number of skills and factors.

Students develop these skills only if genuine mathematical problem solving is taking place. The solved examples in the textbooks give the idea that problem solving is a linear process with no false starts or illogical attempts. Also, the way the solution is presented does not show how much time and effort are needed to come up with a useful solution. Even the problems suggested at the end of a chapter are usually of the same type as those already solved.

Students may start to practice on less demanding problems. However, as they become more confident, problems should be varied and more demanding. Students need to understand that even when no solution to the problem is obtained, learning is taking place. It is the time and effort spent on finding a solution that contribute to the learning process. All the thinking taking place and the organization of one's thoughts during the problem-solving process contribute directly to the learning process.

Key Conclusions

  1. Problem solving is non-linear. Real problem solving involves false starts, revisiting, and revision — unlike the clean solutions in textbooks.
  2. Effort is itself learning. Time spent struggling with a problem is valuable even when no final solution is reached.
  3. Progressively harder problems. Students should begin with manageable problems and gradually encounter more varied and demanding ones.
  4. Skills must be explicitly taught. Concepts, thinking, reasoning, attitude, and metacognition do not develop automatically — they require deliberate instruction.
  5. Design problems for specific skills. Problems can be crafted to focus on particular skills (e.g., rate of work, critical thinking, metacognition).

Summary of Skills Covered in Parts 1–7

Across the seven parts of this research paper, the following skill domains have been examined:

Part Skill Domain Key Insight
1–2Concepts and factsDeep conceptual understanding is required; memorizing formulas is insufficient.
3Basic mathematical skillsAlgebraic manipulation and reading comprehension are prerequisites.
4Teaching concepts through problemsRate of work and exponential growth illustrate progressive skill building.
5Critical thinking and reasoningInductive and deductive reasoning must both be developed; open-ended problems help.
6Attitude and metacognitionPerseverance and self-monitoring are as important as procedural skill.
7Group work and cooperationCollaborative problem solving develops communication and multiplies approaches.

Acknowledgement

The author would like to thank all colleagues at UGRU with whom he had fruitful discussions about this work.

References

  1. [1] D. M. Burton, "The History of Mathematics, an Introduction", McGraw Hill, 2007.
  2. [2] G. Stanic and J. Kilpatrick, "Historical Perspective on Problem Solving in the Mathematics Curriculum", In R. Charles and E. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1-22), Reston, VA: National Council of Teachers of Mathematics, 1989.
  3. [3] K. Stacey, "The Place of Problem Solving in Contemporary Mathematics Curriculum Documents", The Journal of Mathematical Behavior, 24, pp. 341, 2005.
  4. [4] P. Halmos, "The Heart of Mathematics", American Mathematical Monthly, 87 (1980) (7), pp. 519-524.
  5. [5] W. H. Cockcroft, "Mathematics Counts". Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office, (ED.) 1982.
  6. [6] R. E. Reys, M. M. Lindquist, D. V. Lambdin, N. L. Smith & M. N. Suydam, "Helping Children Learn Mathematics", (6th ed.), John Wiley & Sons, Inc., New York, 2001.
  7. [7] S. D. Schafersman, "An Introduction to Critical Thinking", http://www.freeinquiry.com/critical-thinking.html
  8. [8] A. H. Schoenfeld, "Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-Making in Mathematics", Handbook for Research on Mathematics, Edited by D. D. Grouws, MacMillan, New York, 1992.
  9. [9] H. L. Schoen and R. I. Charles, "Teaching Mathematics Through Problem Solving", NCTM catalog, 2003.
  10. [10] G. Polya, "How to Solve It: A New Aspect of Mathematical Method", Second edition, Penguin Books, 1990.
  11. [11] P. S. Wilson, "Research Ideas For the Classroom: High School Mathematics", Chapter 4, MacMillan, 1993.
  12. [12] A. Dendane, 7th Annual Research Conference, "Problem Based Learning in UGRU", UAE University, Al Ain, UAE, April 2006.
  13. [13] M. Dugopolski, J. Coburn & A. G. Bluman, "Algebra for College Students", McGraw Hill, 2006.
  14. [14] J. Metcalfe & A. P. Shimamura, "Metacognition: Knowing about Knowing", Cambridge, MA: MIT Press, 1994.
  15. [15] R. J. Swartz, A. L. Costa, B. K. Beyer, R. Reagan and B. Kallick, "Thinking Based Learning: Activating Students' Potential", Christopher-Gordon Publishers, Norwood, MA, USA, 2007.
  16. [16] R. J. Swartz, "Infusing the Teaching of Thinking Into Content Instruction", A. L. Costa (Ed), Developing Minds: A Resource Book for Teaching Thinking, Vol. 1, Alexandria, Virginia, 1991.
  17. [17] R. J. Swartz and D. N. Perkins, "Teaching Thinking: Issues and Approaches", Cheltenham, Australia: Hawker Brownlow Education, 1990.
  18. [18] D. Gough, "Thinking About Thinking", VA: National Association of Elementary School Principals, Alexandria, 1991.
  19. [19] S. L. Kong, "Using Thinking Skills to Intrinsically Motivate Effective Learning: An Observation From a Teacher Education Classroom", The Korean Journal of Thinking and Problem Solving, 16(1), pp. 75-89, 2006.
  20. [20] B. Kramarski & Z. R. Mevarech, "Enhancing Mathematical Reasoning in the Classroom: The Effects of Cooperative Learning and Metacognitive Training", American Educational Research Journal, Vol. 40, No. 1, 281-310 (2003).

Appendix A: Solution to Problem 6

We first rewrite the equations obtained for the two boats crossing the river:

\[ S_1 t_1 = 1400 \quad (1) \] \[ 1400 + S_2 t_1 = X \quad (2) \] \[ S_1 t_2 = X + 600 \quad (3) \] \[ S_2 t_2 = 2X - 600 \quad (4) \]

The question in Problem 6 is to find the width of the river \( X \) and not necessarily all the other unknowns. We use equations (1) and (2) to express the speed of both boats in terms of \( t_1 \):

\[ S_1 = \frac{1400}{t_1} \quad (5) \] \[ S_2 = \frac{X - 1400}{t_1} \quad (6) \]

We next substitute \( S_1 \) and \( S_2 \) in equations (3) and (4) with their expressions in (5) and (6) to write:

\[ \left( \frac{1400}{t_1} \right) t_2 = X + 600 \quad (7) \] \[ \left( \frac{X - 1400}{t_1} \right) t_2 = 2X - 600 \quad (8) \]

Let the ratio of times be \( T = \frac{t_2}{t_1} \). This allows us to write equations (7) and (8) as follows:

\[ 1400 T = X + 600 \quad (9) \] \[ (X - 1400)T = 2X - 600 \quad (10) \]

We now end up with two equations with two unknowns (\( X \) and \( T \)) that can easily be solved using substitution. Isolating \( T \) in equation (9) gives:

\[ T = \frac{X + 600}{1400} \]

Substituting this into equation (10):

\[ (X - 1400) \cdot \frac{X + 600}{1400} = 2X - 600 \]

Cross-multiplying:

\[ (X - 1400)(X + 600) = 1400(2X - 600) \]

Expanding both sides:

\[ X^2 + 600X - 1400X - 840000 = 2800X - 840000 \] \[ X^2 - 800X - 840000 = 2800X - 840000 \] \[ X^2 - 3600X = 0 \] \[ X(X - 3600) = 0 \]

Since \( X \gt 0 \), the width of the river is:

\[ \boxed{X = 3600 \text{ meters}} \]

Verification: From equation (9): \( T = (3600 + 600)/1400 = 4200/1400 = 3 \). From equation (10): \( T = (7200 - 600)/(3600 - 1400) = 6600/2200 = 3 \). Both give \( T = 3 \), confirming the solution. ✓

Frequently Asked Questions

Why is group work important for mathematical problem solving?

Group work is important because it combines the strengths of multiple learners, exposes students to different approaches, and develops communication skills. Research shows that cooperative learning combined with metacognitive activities has a positive effect on students' problem-solving abilities. Group work also prepares students for collaborative professional environments.

Why do textbook examples give a misleading picture of problem solving?

Textbook solved examples present problem solving as a linear, clean process with no false starts, dead ends, or failed attempts. In reality, problem solving involves uncertainty, revisiting earlier steps, and significant time and effort. Students need to understand this gap between textbook presentation and genuine problem-solving practice.

Does learning occur even when a problem is not solved?

Yes. The thinking process itself, the organization of ideas, and the effort spent searching for a solution all contribute to learning. Students should understand that the time and effort invested in problem solving are valuable even when no final answer is reached.

What is the solution to Problem 6 (the two-boats river problem)?

The width of the river is 3600 meters. Using the ratio \( T = t_2/t_1 \), the two key equations are \( 1400T = X + 600 \) and \( (X - 1400)T = 2X - 600 \), which combine to give \( X^2 - 3600X = 0 \), so \( X = 3600 \) m.

What skills are needed for mathematical problem solving according to this research?

According to this research, problem solving requires: (1) basic algebraic and arithmetic skills, (2) reading comprehension and information extraction, (3) critical thinking, (4) inductive and deductive reasoning, (5) perseverance and a positive attitude, (6) metacognitive awareness, (7) cooperative learning skills, and (8) genuine engagement with non-routine problems.

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