Skills Needed for Mathematical Problem Solving (1) Paper presented (Dr A Dendane) at the 10th Annual Research Conference - UAE University - 13th -16th April, 2009 ABSTRACT
Mathematical problem solving is one of the most important topics to learn and also one of the most complex to teach. The main goal in teaching mathematical problem solving is that students develop a generic ability to solve real life problems and apply mathematics in real life situations. Problem solving can also be used, as a teaching method, for a deeper understanding of concepts. Successful mathematical problem solving depends upon many factors and skills with different characteristics. In fact one of the main difficulties in learning problem solving is the fact that many skills are needed for a learner to be an effective problem solver. Also, these factors and skills make the teaching of problem solving one of the most complex topic to teach. This paper will discuss the idea that problem solving is a process that needs to be understood by instructors so that they can develop better and more effective classroom activities and tasks. The necessary skills for problem solving as well as the methods and strategies to teach or facilitate them are discussed. 1. INTRODUCTION
Mathematical problem solving related to counting and daily transactions of commerce has been present from the earliest days of human experience. Geometry was also widely used in land measurement [1]. Mathematics is now used to quantify numerically and spatially natural as well as man-made situations. It is used to solve problems and with the advent of the computer technology, it has helped in making social, economical and technological advances that few decades ago would have been unthinkable. Learning mathematical facts and contents is important but is not enough. Students should learn how to use these facts to develop their thinking skills and solve problems. Mathematics educators have accepted the idea that that the development of problem solving ability deserves special attention [2 ] and that one of the most important components in any mathematics curriculum or program is genuine mathematical problem solving [3][4][5]. If well facilitated, mathematical problem solving may help students 1. develop and improve the generic ability to solve real life problems, [6] 2. develop critical thinking skills and reasoning, [7]
3. gain deep understanding of concepts, [8 ][9] 4. work in groups, interact with and help each other. Most of the mathematics textbooks suggest few types of problems as examples with detailed solutions and then suggest similar problems as exercises. Students’ learning is limited if only routine problems are solved. The problems used to create genuine learning opportunities should be of the challenging type and not only those similar to ones already solved in the past. Mathematical problem solving is a process that involves a set of factors and tasks to achieve a defined goal. It depends on many skills and factors which therefore makes it challenging both to learn and to te ach. If the instructor’s understanding of the process is limited, difficulties in teaching mathematical problem solving, will arise. Hence the great need to understand these factors and skills if we want to help our students acquire this important process. In this paper, I am addressing issues and concerns related to mathematical problem solving. However, much of what is said in this paper may be applied to any problem solving of the quantitative type such as those encountered in physics, chemistry, Business and Engineering. Our students, at the UAE University, have difficulties in mathematical problem solving and we need to be aware of it. I believe that we need to address this issue and discuss it if we want to improve the teaching of quantitative problem solving which will greatly improve our students’ thinking skills and problem solving abilities in general. My purpose here is to present some of these issues in order to stimulate discussions. Suggestions on how to facilitate certain aspects of mathematical problem solving, as I apply them in my classes, are also presented. 2. MATHEMATICAL PROBLEM SOLVING AS A PROCESS
Mathematical problem solving is a process depending on many phases. There are four main phases in mathematical problem solving [10]. 1. 2. 3. 4.
Understand the problem Devise a plan Carry out the plan Look back
In solving a problem similar, or close, to another problem solved in the past, we will surely follow a linear process as shown in figure 1. The term linear is used to mean that the learner goes from one phase to another until the problem is solved.
Fig. 1: Four steps in mathematical problem solving. (linear model) However this linear process may not be enough when solving a genuine and challenging problem [11]. It is a possible that we carry out the plan but it does not give the right answer or it does not even work. We therefore need to look back critically at the formulation of the problem as well as the information given in the problem. A more practical model of the process is shown in figure 2. The learner need to understand that in solving a problem he needs to look back and reexamine the whole problem during the problem solving process and be ready to restart again if necessary.
Fig. 2: Four steps and looking back at the problem. (More practical model) One of the difficulties in problem solving is the fact that several steps are needed to solve a given problem. Also at each step, students need to use several skills. There are also many factors and skills that determine the outcome of the process. 3. FACTORS AND SKILLS INVOLVED IN PROBLEM SOLVING
The fact that mathematical problem solving is a process that depends on many skills and factors makes it complex to learn and to teach. Figure 3 shows the most important skills and factors necessary for successful problem solving [3]. Routine problems may not need all the skills listed below, however challenging problems need a strategy involving most if not all the skills and attitudes shown in figure 3. In fact the latest development in the education of mathematics is to teach some of these skills through problem solving [9].
Fig. 3: Factors and skills involved in problem solving. Based on my experience and the experience of others, I will discuss and suggest ways to facilitate some of the most important factors that affect the performance of students when solving a mathematical problem. Facilitating and guiding students is one of the most efficient ways to “teach" problem solving [12]. Facilitating is suitable for creating opportunities where students learn by doing. Mathematical problem solving is a constructivist process where student learn to solve problems by making connections to what they already know. Instructors can guide students to make these connections.
3.1 Concepts and Facts in Problem Solving
Learners need a deep understanding of algebraic, geometric, statistical concepts and facts to solve challenging mathematical problems. Here I will explain how I used a set of problems assigned as homework with my level (2) advanced mathematics classes to reinforce the concept of rate of work that was necessary to solve the problem. Step 1: The concept of rate of work is first presented then discussed. Then the whole class discussed the solution to the following problem. Problem 1: It takes Carla 1 hour longer to mow the lawn than it takes Sharon to mow the lawn. If they can mow the lawn in 5 hours working together, then how long would it take each girl by herself? [13] Step 2: Students were then asked to solve the following problem as a home work, where the same concept as problem 1 is needed. Problem 2:
John takes 3 hours longer than Andrew to peel 500 pounds (lb) of apples. If together they can peel 500 lb of apples in 8 hours, then how long would it take each one working alone? [13] Although the above problem was discussed in class before they attempted to solve it, to make sure that students understood it, they had difficulties in solving it. I decided that the concept of rate of work has to be discussed again. Few days later few students managed to solve the given problem and the solution to problem 2 was discussed with the whole class. Step 3: In order to assess students’ understanding of the concept of the rate of work and the process of problem solving, I assigned the following problem as a home work. Problem 3: It takes pump A 2 hours less time than pump B to empty a certain swimming pool. Pump A is started at 8:00 A.M., and pump B is started at 11:00 A.M. If the pool is still half full at 5:00 P.M., then how long would it take pump A working alone? [13]
Problem 3 also needs a deep understanding of the concept of the rate of work. Except for a few students, most found the problem very challenging and could not solve it. I carefully examined the solutions generated by students and I understood that the concept of rate of work was their main difficulty. I decided to give a full hour lesson on the rate of work with many examples and I made sure that not only they understood the concept but also how to use it to formulate problems. I requested that they look again at the problem. Few days later, more than half the class solved problem 3 correctly.
Step 4: I then assigned the following problem in a quiz. Problem 4:
It takes pump B 2 hours more time than pump A to fill a swimming pool. Both pumps are started at 7 am. At 10 am pump A breaks down. It took 1 hour to repair it and then was restarted again. At 3 pm 80 % of the swimming pool was filled with water. How long would it take each pump working alone to fill the swimming pool? About half the class solved the problem correctly and a quarter of the class had solution with minor mistakes. I carefully examined the solutions generated by students and it was clear that the students had a better understanding of the concept of the rate of work and more importantly they knew how to apply it to solve problems. Students fail to solve problems involving concepts that are not thoroughly understood. Moreover, mathematical problems may be used as teaching methodologies not only to introduce concepts but also to help students gain a deeper understanding of these concepts [9]. In fact some concepts cannot be thoroughly understood unless they are used in problem solving or any other activity where critical thinking and reasoning are involved. Mathematical problem solving may also be used to introduce a new concept. An example of a mathematical problem that may be used to introduce a new concept is now presented. Problem 5: The present population of the UAE is 4.5 million. If we assume that the population grows at an annual rate r = 3% for the next 15 years, what will be the population P of the UAE in t years?(assume t is smaller than 15)
Students can easily be guided to use percentages and come up with the following result. P(t)=4.5(1+3%) t At this point the concept of exponential functions can easily be introduced using the result obtained. Students will have understood that exponential functions may be used to solve population problems and make a connection between the concept and its possible application. Experience shows that students are more motivated when solving problems related to their daily life [12]. 3.2 Skills in Problem Solving
Basic mathematical skills such as solving equations and inequalities are necessary for mathematical problem solving. Formulating a problem can be very demanding but simplifying and solving the equation obtained, for example, is necessary to answer the
question in the problem. Students who cannot manipulate algebraic expressions will definitely have difficulties in problem solving. Our students, at UGRU, have difficulties in reading with understanding and extracting the information from the text of the problem. This skill has to be taught explicitly to our students. It is the first step in problem solving and students cannot make any progress if the problem is not understood. 3.3 Thinking and Reasoning in Problem Solving Critical thinking is needed in all steps of problem solving. Students do not look back critically at the solution of a problem once it is solved. They tend to accept whatever answer they have obtained. Critical thinking is needed when extracting information from the text of the problem, formulating and solving the problem and analyzing the solution obtained. I now give an example of a problem that may be used to give opportunities to students to think critically. Problem 6: Two boats on opposite banks of a river start moving towards each other. They first pass each other 1400 meters from one bank. They each continue to the opposite bank, immediately turn around and start back to the other bank. When they pass each other a second time, they are 600 meters from the other bank. We assume that each boat travels at a constant speed all along the journey. Is it possible to find the width of the river using the given information? Students are guided to make a diagram and write the equations to formulate the problem.(see solution in appendix A) Let t = 0 be the time when the two boats start moving. Let S1 and S2 be the constant speeds of the two boats and t1 the time when they first pass each other and t2 the time when they pass each other for the second time. Let X the width of the river.
Fig. 4: Diagram including unknowns in problem 6. S1t1 = 1400 1400 + S2 t1 = X S1 t2 = X + 600 S2 t2 = 2X - 600 Because there are five unknowns and only four equations in the above system of equations, students are tempted to say that it is not possible to find the width X of the river. This particular problem can be used to generate activities where students will have to think in order to come up with a correct answer to the question in the problem. Enough time should be given to students to discuss such a challenging problem. They should be allowed to work in groups and if students cannot finish the work in class, they should be allowed to continue solving the problem as a homework assignment. Students will not benefit if the solution is given to them without giving them enough time to investigate this type of problems. Learning will take place through the time and efforts spent on finding a solution even if they have difficulties finding one. These types of problems involve a lot of thinking both to understand the problem and to come up with a solution and they are necessary if we want our students to develop their thinking skills to the highest level possible and become genuine problem solvers. It is not difficult to design problems that can activate students thinking. One has to avoid problems with one obvious solution. Note that the question in problem 6 does not say “find the width of the river" but “is it possible to find the width of the river using
the given information?". A question of the type “find the width of the river" already assumes that it is possible to find the width X. However questions of the second type are more demanding and therefore suitable to develop thinking skills.
Open ended problems, usually with many valid answers, can also be used as tools to generate activities where students have to think in order to come with answers. Useful discussions can be generated in order to compare all valid answers generated by students. I now give an example of an open ended problem. Problem 7: Create a set of data points that satisfies the following conditions: • The set includes 8 data values. • The range of the data set is 20. • The median is equal to the mean. Show that your data set satisfies the conditions.
Logical reasoning is fundamental to success in mathematics in general and problem solving in particular. Inductive reasoning uses easy to understand cases to identify patterns and it can also be used to establish relationships between mathematical objects. Deductive reasoning uses mathematical definitions, axioms, rules and theorems to draw conclusions. Deductive reasoning is fundamental to critical thinking. Geometry problems may be used as tools to develop deductive reasoning [14]. I believe that both must be used and explicitly discussed with students. 3.4 Attitude and Problem Solving
Learners must persevere and take the risk of failing in solving a given problem. They must also understand that learning takes place even when they are not able to solve a problem. What count is the time and efforts spent looking for a solution. Students do not learn much from problems that they can easily solve, they learn more from challenging problems where they have to persevere. However, students left alone with challenging problems to solve may get frustrated and develop a negative attitude towards mathematical problem solving. The instructor’s task is not easy. He must not give too much information so that students will not have anything to think about. But also he should give enough help so that students will not be frustrated [10]. I found that historical facts about mathematics and real life mathematical problems motivate students. I use both to highlight the power of mathematics and its applications. Below is a real life problem related to trigonometry about measuring the circumference of the earth [1]. Problem 8: More than two thousand years ago Eratosthenes of Cyrene (276 BC-194 BC), a Greek mathematician, used the rays from the sun and the shadow of a stick to measure angle α = 72 degrees. Assume th at the ray from the sun falls vertical at Syene and the stick used at Alexandria is also vertical. Assume also that the earth is circular and find the circumference of the earth.
Fig. 5: Rays from the sun and the shadow to measure angle . Students are more interested in real life problems because they make sense and also give reasons to study mathematics. 3.5 Metacognition and Problem Solving
The term metacognition refers to a learners knowledge about his or her process of cognition and the ability to control and monitor those processes as a function of the feedback the learner receives via outcomes of learning [14]. Learners must control and reflect on their cognitive processes in order to solve challenging problems. It is thought that the best way to help students become aware about their own thinking is to create opportunities where they have to explain their thinking explicitly. As an example let us look at the following problem. Problem 9: Ahmed walked at a constant speed of 6 km/hour along a straight line from A to B, then walked back along the same line from B to A at a constant speed of 4 km/hour. What is the average speed over the entire trip? (Definition: Average speed = total distance / total time)
When the above problem was given to students in level (2) advanced mathematics, all students answered the question as follows:
average speed = (6 + 4) / 2 = 5 km/hour
When asked to explain their answer, they said that any average is found by adding all data values and divide the sum obtained by the number of data values. They simply ignored the definition given to them. I next guided them to work in groups, look back at the problem and use the definition to answer the question. They had to justify the steps of their calculations and their thinking within their group. When the correct answer based on the definition was obtained, I asked students to discuss within their groups what happened and why all students answered the question without taking into account the definition. I explicitly ask them to think about the way they solved the problem and why. Metacognitive skills help students to analyze both the question and the solution developed. They also help students to look back at the problem and restart if necessary. In designing activities that may help students be aware of their metacognitive skills, I have taken into account the following: 1. Metacognitive skills have to be taught explicitly.[15-18] 2. Only genuine mathematical problems, that students have not solved before, help them develop metacognitive skills. 3. Students need to explain to other students and the teacher their way of thinking. 4. I sometimes use examples to explain my own thinking in solving problems. 3.6 Group Work and Problem Solving Experience shows that group work is ver y 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’ abiliti es to solve problems [19-20]. Group work also prepares students for the future where they have to work together on large problems and 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 factor s. 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 dem anding 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 cont ribute 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 h ad 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 W iley & 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 Resear ch 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
