What Is A 'Problem-Solving Approach'?
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As the emphasis has shifted from teaching problem solving to teaching via problem solving (Lester, Masingila, Mau, Lambdin, dos Santon and Raymond, 1994), many writers have attempted to clarify what is meant by a problem-solving approach to teaching mathematics. The focus is on teaching mathematical topics through problem-solving contexts and enquiry-oriented environments which are characterised by the teacher 'helping students construct a deep understanding of mathematical ideas and processes by engaging them in doing mathematics: creating, conjecturing, exploring, testing, and verifying' (Lester et al., 1994, p.154). Specific characteristics of a problem-solving approach include:
- interactions between students/students and teacher/students (Van Zoest et al., 1994)
- mathematical dialogue and consensus between students (Van Zoest et al., 1994)
- teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991)
- teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991)
- teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994)
- teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994)
- A further characteristic is that a problem-solving approach can be used to encourage students to make generalisations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).
Schoenfeld (in Olkin and Schoenfeld, 1994, p.43) described the way in which the use of problem solving in his teaching has changed since the 1970s:
My early problem-solving courses focused on problems amenable to solutions by Polya-type heuristics: draw a diagram, examine special cases or analogies, specialize, generalize, and so on. Over the years the courses evolved to the point where they focused less on heuristics per se and more on introducing students to fundamental ideas: the importance of mathematical reasoning and proof..., for example, and of sustained mathematical investigations (where my problems served as starting points for serious explorations, rather than tasks to be completed).
Schoenfeld also suggested that a good problem should be one which can be extended to lead to mathematical explorations and generalisations. He described three characteristics of mathematical thinking:
- valuing the processes of mathematization and abstraction and having the predilection to apply them
- developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense-making (Schoenfeld, 1994, p.60).
- As Cobb et al. (1991) suggested, the purpose for engaging in problem solving is not just to solve specific problems, but to 'encourage the interiorization and reorganization of the involved schemes as a result of the activity' (p.187). Not only does this approach develop students' confidence in their own ability to think mathematically (Schifter and Fosnot, 1993), it is a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others (NCTM, 1989). Because it has become so predominant a requirement of teaching, it is important to consider the processes themselves in more detail.
The Role of Problem Solving in Teaching Mathematics as a Process
Problem solving is an important component of mathematics education because it is the single vehicle which seems to be able to achieve at school level all three of the values of mathematics listed at the outset of this article: functional, logical and aesthetic. Let us consider how problem solving is a useful medium for each of these.
It has already been pointed out that mathematics is an essential discipline because of its practical role to the individual and society. Through a problem-solving approach, this aspect of mathematics can be developed. Presenting a problem and developing the skills needed to solve that problem is more motivational than teaching the skills without a context. Such motivation gives problem solving special value as a vehicle for learning new concepts and skills or the reinforcement of skills already acquired (Stanic and Kilpatrick, 1989, NCTM, 1989). Approaching mathematics through problem solving can create a context which simulates real life and therefore justifies the mathematics rather than treating it as an end in itself. The National Council of Teachers of Mathematics (NCTM, 1980) recommended that problem solving be the focus of mathematics teaching because, they say, it encompasses skills and functions which are an important part of everyday life. Furthermore it can help people to adapt to changes and unexpected problems in their careers and other aspects of their lives. More recently the Council endorsed this recommendation (NCTM, 1989) with the statement that problem solving should underly all aspects of mathematics teaching in order to give students experience of the power of mathematics in the world around them. They see problem solving as a vehicle for students to construct, evaluate and refine their own theories about mathematics and the theories of others.
According to Resnick (1987) a problem-solving approach contributes to the practical use of mathematics by helping people to develop the facility to be adaptable when, for instance, technology breaks down. It can thus also help people to transfer into new work environments at this time when most are likely to be faced with several career changes during a working lifetime (NCTM, 1989). Resnick expressed the belief that 'school should focus its efforts on preparing people to be good adaptive learners, so that they can perform effectively when situations are unpredictable and task demands change' (p.18). Cockcroft (1982) also advocated problem solving as a means of developing mathematical thinking as a tool for daily living, saying that problem-solving ability lies 'at the heart of mathematics' (p.73) because it is the means by which mathematics can be applied to a variety of unfamiliar situations.
Problem solving is, however, more than a vehicle for teaching and reinforcing mathematical knowledge and helping to meet everyday challenges. It is also a skill which can enhance logical reasoning. Individuals can no longer function optimally in society by just knowing the rules to follow to obtain a correct answer. They also need to be able to decide through a process of logical deduction what algorithm, if any, a situation requires, and sometimes need to be able to develop their own rules in a situation where an algorithm cannot be directly applied. For these reasons problem solving can be developed as a valuable skill in itself, a way of thinking (NCTM, 1989), rather than just as the means to an end of finding the correct answer.
Many writers have emphasised the importance of problem solving as a means of developing the logical thinking aspect of mathematics. 'If education fails to contribute to the development of the intelligence, it is obviously incomplete. Yet intelligence is essentially the ability to solve problems: everyday problems, personal problems ... '(Polya, 1980, p.1). Modern definitions of intelligence (Gardner, 1985) talk about practical intelligence which enables 'the individual to resolve genuine problems or difficulties that he or she encounters' (p.60) and also encourages the individual to find or create problems 'thereby laying the groundwork for the acquisition of new knowledge' (p.85). As was pointed out earlier, standard mathematics, with the emphasis on the acquisition of knowledge, does not necessarily cater for these needs. Resnick (1987) described the discrepancies which exist between the algorithmic approaches taught in schools and the 'invented' strategies which most people use in the workforce in order to solve practical problems which do not always fit neatly into a taught algorithm. As she says, most people have developed 'rules of thumb' for calculating, for example, quantities, discounts or the amount of change they should give, and these rarely involve standard algorithms. Training in problem-solving techniques equips people more readily with the ability to adapt to such situations.
A further reason why a problem-solving approach is valuable is as an aesthetic form. Problem solving allows the student to experience a range of emotions associated with various stages in the solution process. Mathematicians who successfully solve problems say that the experience of having done so contributes to an appreciation for the 'power and beauty of mathematics' (NCTM, 1989, p.77), the 'joy of banging your head against a mathematical wall, and then discovering that there might be ways of either going around or over that wall' (Olkin and Schoenfeld, 1994, p.43). They also speak of the willingness or even desire to engage with a task for a length of time which causes the task to cease being a 'puzzle' and allows it to become a problem. However, although it is this engagement which initially motivates the solver to pursue a problem, it is still necessary for certain techniques to be available for the involvement to continue successfully. Hence more needs to be understood about what these techniques are and how they can best be made available.
In the past decade it has been suggested that problem-solving techniques can be made available most effectively through making problem solving the focus of the mathematics curriculum. Although mathematical problems have traditionally been a part of the mathematics curriculum, it has been only comparatively recently that problem solving has come to be regarded as an important medium for teaching and learning mathematics (Stanic and Kilpatrick, 1989). In the past problem solving had a place in the mathematics classroom, but it was usually used in a token way as a starting point to obtain a single correct answer, usually by following a single 'correct' procedure. More recently, however, professional organisations such as the National Council of Teachers of Mathematics (NCTM, 1980 and 1989) have recommended that the mathematics curriculum should be organized around problem solving, focusing on:
- developing skills and the ability to apply these skills to unfamiliar situations
- gathering, organising, interpreting and communicating information
- formulating key questions, analyzing and conceptualizing problems, defining problems and goals, discovering patterns and similarities, seeking out appropriate data, experimenting, transferring skills and strategies to new situations
- developing curiosity, confidence and open-mindedness (NCTM, 1980, pp.2-3).
One of the aims of teaching through problem solving is to encourage students to refine and build onto their own processes over a period of time as their experiences allow them to discard some ideas and become aware of further possibilities (Carpenter, 1989). As well as developing knowledge, the students are also developing an understanding of when it is appropriate to use particular strategies. Through using this approach the emphasis is on making the students more responsible for their own learning rather than letting them feel that the algorithms they use are the inventions of some external and unknown 'expert'. There is considerable importance placed on exploratory activities, observation and discovery, and trial and error. Students need to develop their own theories, test them, test the theories of others, discard them if they are not consistent, and try something else (NCTM, 1989). Students can become even more involved in problem solving by formulating and solving their own problems, or by rewriting problems in their own words in order to facilitate understanding. It is of particular importance to note that they are encouraged to discuss the processes which they are undertaking, in order to improve understanding, gain new insights into the problem and communicate their ideas (Thompson, 1985, Stacey and Groves, 1985).
Conclusion
It has been suggested in this chapter that there are many reasons why a problem-solving approach can contribute significantly to the outcomes of a mathematics education. Not only is it a vehicle for developing logical thinking, it can provide students with a context for learning mathematical knowledge, it can enhance transfer of skills to unfamiliar situations and it is an aesthetic form in itself. A problem-solving approach can provide a vehicle for students to construct their own ideas about mathematics and to take responsibility for their own learning. There is little doubt that the mathematics program can be enhanced by the establishment of an environment in which students are exposed to teaching via problem solving, as opposed to more traditional models of teaching about problem solving. The challenge for teachers, at all levels, is to develop the process of mathematical thinking alongside the knowledge and to seek opportunities to present even routine mathematics tasks in problem-solving contexts.
References
Carpenter, T. P. (1989). 'Teaching as problem solving'. In R.I.Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.187-202). USA: National Council of Teachers of Mathematics.
Clarke, D. and McDonough, A. (1989). 'The problems of the problem solving classroom', The Australian Mathematics Teacher, 45, 3, 20-24.
Cobb, P., Wood, T. and Yackel, E. (1991). 'A constructivist approach to second grade mathematics'. In von Glaserfield, E. (Ed.), Radical Constructivism in Mathematics Education, pp. 157-176. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Cockcroft, W.H. (Ed.) (1982). Mathematics Counts. Report of the Committee of Inquiry into the Teaching of Mathematics in Schools, London: Her Majesty's Stationery Office.
Evan, R. and Lappin, G. (1994). 'Constructing meaningful understanding of mathematics content', in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 128-143. Reston, Virginia: NCTM.
Gardner, Howard (1985). Frames of Mind. N.Y: Basic Books.
Lester, F.K.Jr., Masingila, J.O., Mau, S.T., Lambdin, D.V., dos Santon, V.M. and Raymond, A.M. (1994). 'Learning how to teach via problem solving'. in Aichele, D. and Coxford, A. (Eds.) Professional Development for Teachers of Mathematics , pp. 152-166. Reston, Virginia: NCTM.
National Council of Teachers of Mathematics (NCTM) (1980). An Agenda for Action: Recommendations for School Mathematics of the 1980s, Reston, Virginia: NCTM.
National Council of Teachers of Mathematics (NCTM) (1989). Curriculum and Evaluation Standards for School Mathematics, Reston, Virginia: NCTM.
Olkin, I. & Schoenfeld, A. (1994). A discussion of Bruce Reznick's chapter. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 39-51). Hillsdale, NJ: Lawrence Erlbaum Associates.
Polya, G. (1980). 'On solving mathematical problems in high school'. In S. Krulik (Ed). Problem Solving in School Mathematics, (pp.1-2). Reston, Virginia: NCTM.
Resnick, L. B. (1987). 'Learning in school and out', Educational Researcher, 16, 13-20..
Romberg, T. (1994). Classroom instruction that fosters mathematical thinking and problem solving: connections between theory and practice. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 287-304). Hillsdale, NJ: Lawrence Erlbaum Associates.
Schifter, D. and Fosnot, C. (1993). Reconstructing Mathematics Education. NY: Teachers College Press.
Schoenfeld, A. (1994). Reflections on doing and teaching mathematics. In A. Schoenfeld (Ed.). Mathematical Thinking and Problem Solving. (pp. 53-69). Hillsdale, NJ: Lawrence Erlbaum Associates.
Stacey, K. and Groves, S. (1985). Strategies for Problem Solving, Melbourne, Victoria: VICTRACC.
Stanic, G. and Kilpatrick, J. (1989). 'Historical perspectives on problem solving in the mathematics curriculum'. In R.I. Charles and E.A. Silver (Eds), The Teaching and Assessing of Mathematical Problem Solving, (pp.1-22). USA: National Council of Teachers of Mathematics.
Swafford, J.O. (1995). 'Teacher preparation'. in Carl, I.M. (Ed.) Prospects for School Mathematics , pp. 157-174. Reston, Virginia: NCTM.
Swafford, J.O. (1995). 'Teacher preparation'. in Carl, I.M. (Ed.) Prospects for School Mathematics , pp. 157-174. Reston, Virginia: NCTM.
Thompson, P. W. (1985). 'Experience, problem solving, and learning mathematics: considerations in developing mathematics curricula'. In E.A. Silver (Ed.), Teaching and Learning Mathematical Problem Solving: Multiple Research Perspectives, (pp.189-236). Hillsdale, N.J: Lawrence Erlbaum.
Van Zoest, L., Jones, G. and Thornton, C. (1994). 'Beliefs about mathematics teaching held by pre-service teachers involved in a first grade mentorship program'. Mathematics Education Research Journal. 6(1): 37-55.
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10
WORD PROBLEMS require practice in translating verbal language into algebraic language. See Lesson 1, Problem 8. Yet, word problems fall into distinct types. Below are some examples.
Example 1. ax±b = c. All problems like the following lead eventually to an equation in that simple form.
Jane spent $42 for shoes. This was $14 less than twice what she spent for a blouse. How much was the blouse?
Solution. Every word problem has an unknown number. In this problem, it is the price of the blouse. Always let x represent the unknown number. That is, let x answer the question.
Let x, then, be how much she spent for the blouse. The problem states that 'This' -- that is, $42 -- was $14 less than two times x.
Here is the equation:
| 2x − 14 | = | 42. |
| 2x | = | 42 + 14 (Lesson 9) |
| = | 56. | |
| x | = | 56 2 |
| = | 28. |
The blouse cost $28.
Example 2. There are b boys in the class. This is three more than four times the number of girls. How many girls are in the class?
Solution. Again, let x represent the unknown number that you are asked to find: Let x be the number of girls.
Algebra Math Problems Solver
(Although b is not known -- it is an arbitrary constant -- it is not what you are asked to find.)
The problem states that 'This' -- b -- is three more than four times x:
| 4x + 3 | = | b. | ||
| Therefore, | ||||
| 4x | = | b − 3 | ||
| x | = | b − 3 4 | . | |
The solution here is not a number, because it will depend on the value of b. This is a type of 'literal' equation, which is very common in algebra.
Example 3. The whole is equal to the sum of the parts.
The sum of two numbers is 84, and one of them is 12 more than the other. What are the two numbers?
Solution. In this problem, we are asked to find two numbers. Therefore, we must let x be one of them. Let x, then, be the first number.
We are told that the other number is 12 more, x + 12.
The problem states that their sum is 84:
= 84
The line over x + 12 is a grouping symbol called a vinculum. It saves us writing parentheses.
We have:
| 2x | = | 84 − 12 |
| = | 72. | |
| x | = | 72 2 |
| = | 36. |
This is the first number. Therefore the other number is
x + 12 = 36 + 12 = 48.
The sum of 36 + 48 is 84.
Example 4. The sum of two consecutive numbers is 37. What are they?
Solution. Two consecutive numbers are like 8 and 9, or 51 and 52.
Let x, then, be the first number. Then the number after it is x + 1.
The problem states that their sum is 37:
= 37
| 2x | = | 37 − 1 |
| = | 36. | |
| x | = | 36 2 |
| = | 18. |
The two numbers are 18 and 19.
Example 5. One number is 10 more than another. The sum of twice the smaller plus three times the larger, is 55. What are the two numbers?
Solution. Let x be the smaller number.
Then the larger number is 10 more: x + 10.
The problem states:
| 2x + 3(x + 10) | = | 55. |
| That implies | ||
| 2x + 3x + 30 | = | 55. Lesson 14. |
| 5x | = | 55 − 30 = 25. |
| x | = | 5. |
That's the smaller number. The larger number is 10 more: 15.
Example 6. Divide $80 among three people so that the second will have twice as much as the first, and the third will have $5 less than the second.
Solution. Again, we are asked to find more than one number. We must begin by letting x be how much the first person gets.
Then the second gets twice as much, 2x.
And the third gets $5 less than that, 2x − 5.
Their sum is $80:
| 5x | = | 80 + 5 |
| x | = | 85 5 |
| = | 17. |
This is how much the first person gets. Therefore the second gets
| 2x | = | 34. |
| And the third gets | ||
| 2x − 5 | = | 29. |
The sum of 17, 34, and 29 is in fact 80.
Example 7. Odd numbers. The sum of two consecutive odd numbers is 52. What are the two odd numbers?
Solution. First, an even number is a multiple of 2: 2, 4, 6, 8, and so on. It is conventional in algebra to represent an even number as 2n, where, by calling the variable 'n,' it is understood that n will take whole number values: n = 0, 1, 2, 3, 4, and so on.
An odd number is 1 more (or 1 less) than an even number. And so we represent an odd number as 2n + 1.
Let 2n + 1, then, be the first odd number. Then the next one will be 2 more -- it will be 2n + 3. The problem states that their sum is 52:
We will now solve that equation for n, and then replace the solution in 2n + 1 to find the first odd number. We have:
| 4n + 4 | = | 52 |
| 4n | = | 48 |
| n | = | 12. |
Therefore the first odd number is 2 · 12 + 1 = 25. And so the next one is 27. Their sum is 52.
Problems
Problem 1. Julie has $50, which is eight dollars more than twice what John has. How much has John? (Compare Example 1.)
First, what will you let x represent?
To see the answer, pass your mouse over the colored area.
To cover the answer again, click 'Refresh' ('Reload').
Do the problem yourself first!
The unknown number -- which is how much that John has.
What is the equation?
2x + 8 = 50.
Here is the solution:
x = $21
Problem 2. Carlotta spent $35 at the market. This was seven dollars less than three times what she spent at the bookstore; how much did she spend there?
Here is the equation.
3x − 7 = 35
Here is the solution:
x = $14
Problem 3. There are b black marbles. This is four more than twice the number of red marbles. How many red marbles are there? (Compare Example 2.)
Here is the equation.
2x + 4 = b
Here is the solution:
| x = | b − 4 2 |
Problem 4. Janet spent $100 on books. This was k dollars less than five times what she spent on lunch. How much did she spend on lunch?
Here is the equation.
5x − k = 100
Here is the solution:
| x = | 100 + k 5 |
Problem 5. The whole is equal to the sum of the parts.
The sum of two numbers is 99, and one of them is 17 more than the other. What are the two numbers? (Compare Example 3.)
Here is the equation.
Here is the solution:
Problem 6. A class of 50 students is divided into two groups; one group has eight less than the other; how many are in each group?
Here is the equation.
Here is the solution:
| x | = | 29 |
| x − 8 | = | 21 |
Problem 7. The sum of two numbers is 72, and one of them is five times the other; what are the two numbers?
Here is the equation.
x + 5x = 72.
Here is the solution:
x = 12. 5x = 60.
Problem 8. The sum of three consecutive numbers is 87; what are they? (Compare Example 4.)
Here is the equation.
Here is the solution:
28, 29, 30.
Problem 9. A group of 266 persons consists of men, women, and children. There are four times as many men as children, and twice as many women as children. How many of each are there?
(What will you let x equal -- the number of men, women, or children?)
| Let x | = | The number of children. Then |
| 4x | = | The number of men. And |
| 2x | = | The number of women. |
| Here is the equation: | ||
x + 4x + 2x = 266
Here is the solution:
x = 38. 4x = 152. 2x = 76.
Problem 10. Divide $79 among three people so that the second will have three times more than the first, and the third will have two dollars more than the second. (Compare Example 6.)
Here is the equation.
Here is the solution:
$11, $33, $35.
Problem 11. Divide $15.20 among three people so that the second will have one dollar more than the first, and the third will have $2.70 more than the second.
Here is the equation.
Here is the solution:
$3.50, $4.50, $7.20.
Problem 12. Two consecutive odd numbers are such that three times the first is 5 more than twice the second. What are those two odd numbers?
(See Example 7, where we represent an odd number as 2n + 1.)
Solution. Let the first odd number be 2n + 1.
Then the next one is 2n + 3 -- because it will be 2 more.
The problem states—that is, the equation is:
| 3(2n + 1) | = | 2(2n + 3) + 5. | |
| That implies: | |||
| 6n + 3 | = | 4n + 6 + 5. | |
| 2n | = | 8. | |
| n | = | 4. | |
Algebra Iimath Problem Solving
Therefore the first odd number is 2 · 4 + 1 = 9. The next one is 11.
And that is the true solution, because according to the problem:
3 · 9 = 2 · 11 + 5.
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