# Synthesizing major concepts of math reflective paper

Fernandez, and Nelda Hadaway Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime. Problem solving has a special importance in the study of mathematics. A primary goal of mathematics teaching and learning is to develop the ability to solve a wide variety of complex mathematics problems.

Stanic and Kilpatrick 43 traced the role of problem solving in school mathematics and illustrated a rich history of the topic. To many mathematically literate people, mathematics is synonymous with solving problems -- doing word problems, creating patterns, interpreting figures, developing geometric constructions, proving theorems, etc.

On the other hand, persons not enthralled with mathematics may describe any mathematics activity as problem solving.

Learning to solve problems is the principal reason for studying mathematics. National Council of Supervisors of Mathematics 22 When two people talk about mathematics problem solving, they may not be talking about the same thing. The rhetoric of problem solving has been so pervasive in the mathematics education of the 1980s and 1990s that creative speakers and writers can put a twist on whatever topic or activity they have in mind to call it problem solving!

Every exercise of problem solving research has gone through some agony of defining mathematics problem solving. Yet, words sometimes fail. Most people resort to a few examples and a few nonexamples. Reitman 29 defined a problem as when you have been given the description of something but do not yet have anything that satisfies that description.

Reitman's discussion described a synthesizing major concepts of math reflective paper solver as a person perceiving and accepting a goal without an immediate means of reaching the goal. Henderson and Pingry 11 wrote that to be problem solving there must be a goal, a blocking of that goal for the individual, and acceptance of that goal by the individual.

What is a problem for one student may not be a problem for another -- either because there is no blocking or no acceptance of the goal. Schoenfeld 33 also pointed out that defining what is a problem is always relative to the individual. Assume there is a single recording and the Outer beginning groove is 5. The recording plays for 23 minutes.

Mathematics teachers talk about, write about, and act upon, many different ideas under the heading of problem solving. Some have in mind primarily the selection and presentation of "good" problems to students.

Some think of mathematics program goals in which the curriculum is structured around problem content.

### Algorithms

Others think of program goals in which the strategies and techniques of problem solving are emphasized. Some discuss mathematics problem solving in the context of a method of teaching, i. Indeed, discussions of mathematics problem solving often combine and blend several of these ideas.

In this chapter, we want to review and discuss the research on how students in secondary schools can develop the ability to solve a wide variety of complex problems.

## Research on Problem Solving

We will also address how instruction can best develop this ability. A fundamental goal of all instruction is to develop skills, knowledge, and abilities that transfer to tasks not explicitly covered in the curriculum.

Should instruction emphasize the particular problem solving techniques or strategies unique to each task? Will problem solving be enhanced by providing instruction that demonstrates or develops problem solving techniques or strategies useful in many tasks?

We are particularly interested in tasks that require mathematical thinking 34 or higher order thinking skills 17. Throughout the chapter, we have chosen to separate and delineate aspects of mathematics problem solving when in fact the separations are pretty fuzzy for any of us.

Although this chapter deals with problem solving research at the secondary level, there is a growing body of research focused on young children's solutions to word problems 6,30. Readers should also consult the problem solving chapters in the Elementary and Middle School volumes.

Research on Problem Solving Educational research is conducted within a variety of constraints -- isolation of variables, availability of subjects, limitations of research procedures, availability of resources, and balancing of priorities. Synthesizing major concepts of math reflective paper research methodologies are used in mathematics education research including a clinical approach that is frequently used to study problem solving.

Typically, mathematical tasks or problem situations are devised, and students are studied as they perform the tasks. Often they are asked to talk aloud while working or they are interviewed and asked to reflect on their experience and especially their thinking processes.

Waters 48 discusses the advantages and disadvantages of four different methods of measuring strategy use involving a clinical approach.

Schoenfeld 32 describes how a clinical approach may be used with pairs of students in an interview. He indicates that "dialog between students often serves to make managerial decisions overt, whereas such decisions are rarely overt in single student protocols. The basis for most mathematics problem solving research for secondary school students in the past 31 years can be found in the writings of Polya 26,27,28the field of cognitive psychology, and specifically in cognitive science.

Cognitive psychologists and cognitive scientists seek to develop or validate theories of human learning 9 whereas mathematics educators seek to understand how their students interact with mathematics 33,40. The area of cognitive science has particularly relied on computer simulations of problem solving 25,50. If synthesizing major concepts of math reflective paper computer program generates a sequence of behaviors similar to the sequence for human subjects, then that program is a model or theory of the behavior.

Newell and Simon 25Larkin 18and Bobrow 2 have provided simulations of mathematical problem solving. These simulations may be used to better understand mathematics problem solving. Constructivist theories have received considerable acceptance in mathematics education in recent years. In the constructivist perspective, the learner must be actively involved in the construction of one's own knowledge rather than passively receiving knowledge. The teacher's responsibility is to arrange situations and contexts within which the learner constructs appropriate knowledge 45,48.

Even though the constructivist view of mathematics learning is appealing and the theory has formed the basis for many studies at the elementary level, research at the secondary level is lacking.

Our review has not uncovered problem solving research at the secondary level that has its basis in a constructivist perspective. However, constructivism is consistent with current cognitive theories of problem solving and mathematical views of problem solving involving exploration, pattern finding, and mathematical thinking 36,15,20 ; thus we urge that teachers and teacher educators become familiar with constructivist views and evaluate these views for restructuring their approaches to teaching, learning, and research dealing with problem solving.

A Framework It is useful to develop a framework to think about the processes involved in mathematics problem solving.

- Educational Studies in Mathematics, 12 2 , 235-265.
- What are the dimensions of each and what is the area?
- The approximation of roots of equations can be made operational with a calculator or computer to carry out the iteration. Notice that the waveform appears periodic though it is not a pure tone.
- Problem solving in mathematics. Will problem solving be enhanced by providing instruction that demonstrates or develops problem solving techniques or strategies useful in many tasks?

Most formulations of a problem solving framework in U. However, it is important to note that Polya's "stages" were more flexible than the "steps" often delineated in textbooks. These stages were described as understanding the problem, making a plan, carrying out the plan, and looking back. To Polya 28problem solving was a major theme of doing mathematics and "teaching students to think" was of primary importance.

However, care must be taken so that efforts to teach students "how to think" in mathematics problem solving do not get transformed into teaching "what to think" or "what to do. Clearly, the linear nature of the models used in numerous textbooks does not promote the spirit of Polya's stages and his goal of teaching students to think. By their nature, all of these traditional models have the following defects: They depict problem solving as a linear process.

They present problem solving as a series of steps. They imply that solving mathematics problems is a procedure to be memorized, practiced, and habituated. They lead to an emphasis on answer getting. These linear formulations are not very consistent with genuine problem solving activity. They may, however, be consistent with how experienced problem solvers present their solutions and answers after the problem synthesizing major concepts of math reflective paper is completed.

In an analogous way, mathematicians present their proofs in very concise terms, but the most elegant of proofs may fail to convey the dynamic inquiry that went on in constructing the proof. Synthesizing major concepts of math reflective paper aspect of problem solving that is seldom included in textbooks is problem posing, or problem formulation. Although there has been little research in this area, this activity has been gaining considerable attention in U. Brown and Walter 3 have provided the major work on problem posing.

Indeed, the examples and strategies they illustrate show a powerful and dynamic side to problem posing activities. Polya 26 did not talk specifically about problem posing, but much of the spirit and format of problem posing is included in his illustrations of looking back. A framework is needed that emphasizes the dynamic and cyclic nature of genuine problem solving.

A student may begin with a problem and engage in thought and activity to understand it. The student attempts to make a plan and in the process may discover a need to understand the problem better.

Or when a plan has been formed, the student may attempt to carry it out and be unable to do so. The next activity may be attempting to make a new plan, or going back to develop a new understanding of the problem, or posing a new possibly related problem to work on. The framework in Figure 2 is useful for illustrating the dynamic, cyclic interpretation of Polya's 26 stages. It has been used in a mathematics problem solving course at the University of Georgia for many years.

Any of the arrows could describe student activity thought in the process of solving mathematics problems. Clearly, genuine problem solving experiences in mathematics can not be captured by the outer, one-directional arrows alone. It is not a theoretical model. Rather, it is a framework for discussing various pedagogical, curricular, instructional, and learning issues involved with the goals of mathematical problem solving in our schools. Problem solving abilities, beliefs, attitudes, and performance develop in contexts 36 and those contexts must be studied as well as specific problem solving activities.

We have chosen to organize the remainder of this chapter around the topics of problem solving as a process, problem solving as an instructional goal, problem solving as an instructional method, beliefs about problem solving, evaluation of problem solving, and technology and problem solving. Problem Solving as a Process Garofola and Lester 10 have suggested that students are largely unaware of the processes involved in problem solving and that addressing this issue within problem solving instruction may be important.

We will discuss various areas of research pertaining to the process of problem solving. Domain Specific Knowledge To become a good problem solver in mathematics, one must develop a base of mathematics knowledge. How effective one is in organizing that knowledge also contributes to successful problem solving. Kantowski 13 found that those students with a good knowledge base were most able to use the heuristics in geometry instruction.

### Domain Specific Knowledge

Schoenfeld and Herrmann 38 found that novices attended to surface features of problems whereas experts categorized problems on the basis of the fundamental principles involved. Silver 39 found that successful problem solvers were more likely to categorize math problems on the basis of their underlying similarities in mathematical structure. Wilson 50 found that general heuristics had utility only when preceded by task specific heuristics.

The task specific heuristics were often specific to the problem domain, such as the tactic most students develop in working with trigonometric identities to "convert all expressions to functions of sine and cosine and do algebraic simplification.

## No Static at All: Frequency modulation and music synthesis

Algorithms are important in mathematics and our instruction must develop them but the process of carrying out an algorithm, even a complicated one, is not problem solving. The process of creating an algorithm, however, and generalizing it to a specific set of applications can be problem solving. Thus problem solving can be incorporated into the curriculum by having students create their own algorithms.