Skills Needed for Mathematical Problem Solving (7)

Experience shows that group work is very useful in solving problems in general [12]. When the problem given to students is challenging, students are happy to work in groups. In fact is was shown that cooperative learning and metacognitive activities have positive effects on the students’ abilities to solve problems [19-20]. Group work also prepares students for the future where they have to work together on large problems and projects.

3.6 Group Work and Problem Solving 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 problems solving is taking place. The solved examples in the textbooks give the idea that problem solving is a linear process with no false start or illogical attempts. Also the way the solution is presented does not show how much time and efforts 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 efforts spent on finding that contribute to the learning process. All the thinking taking place and the organization of one’s thoughts during the problem solving process contribute to the learning process.

ACKNOWLEDGEMENT

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

REFERENCES

[1] D. M. Burton, “The History of Mathematics, an Introduction", McGraw Hill, 2007.

[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] K. Stacey, “The Place of Problem Solving in Contemporary Mathematics Curriculum Documents", The Journal of Mathematical Behavior, 24, pp.341, 2005.

[4] P. Halmos, “The heart of Mathematics", American Mathematical Monthly, 87 (1980) (7) , pp. 519-524.

[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] 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] S. D. Schafersman, “An Introduction to Critical Thinking", http://www.freeinquiry.com/critical-thinking.html

[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] H. L. Schoen and R. I. Charles, “Teaching Mathematics Through Problem Solving", NCTM catalog, 2003.

[10] G. Polya, “How to Solve it: A New Aspect of mathematical Method?", Seocnd edition, Penguin Books, 1990.

[11] P. S. Wilson, “Research Ideas For the Classroom: High School Mathematics", Chapter 4, MacMillan, 1993.

[12] A. Dendane, 7th Annual Research Conference, “Problem Based Learning in UGRU", UAE University, Al Ain, UAE, April 2006.

[13] M. Dugopolski, J. Coburn & A. G. Bluman, “Algebra for College Students", McGraw Hill, 2006.

[14] J. Metcalfe & A.P. Shimamura, “Metacognition: Knowing about Knowing", Cambridge, MA:MIT Press, 1994.

[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] 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, Alexandra, Virginia, 1991.

[17] R. J. Swartz and D. N. Perkins, “Teaching Thinking: Issues and Approaches, Cheltenham, Australia: Hawker Brownlow Education, 1990.

[18] D. Gough, “Thinking About Thinking", VA: National Association of Elementary School Principals, Alexandria, 1991.

[19] S. L. Kong, The Korean Journal of Thinking and Problem Solving, “Using Thinking Skills to Intrinsically Motivate Effective Learning: An Observation From a Teacher Education Classroom", 16(1), pp 75-89, 2006.

[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.

S1 t1 = 1400         (1)

1400 + S2 t1 = X         (2)

S1 t2 = X + 600         (3)

S2 t2 = 2X - 600

The question in problem 6 is to find X and not all the unknowns. We use equations (1) and (2) to write

S1 = 1400 / t1     (5)     and     S2 = (X - 1400) / t1    (6)

We next substitute S1 and S2 in equations (3) and (4) by their expressions in (5) and (6) to write

( 1400 / t1 ) t2 = X + 600        (7)

( (X - 1400) / t1 ) t2 = 2X - 600        (8)

Let T = t2/t1 and write equation (7) and (8) as follows

1400 T = X + 600         (9)

X*T - 1400T = 2X - 600        (10)

We now end up with two equations with two unknowns that can easily be solved to give X = 3600 meters.

1, 2, 3, 4, 5, 6, 7,

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