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physical & psychological effects of abortion

Studies of the long-term risks of induced abortion, such as difficulties with future pregnancies, show that these risks are minimal. A properly done early abortion may even result in a lower risk of certain obstetrical problems with later pregnancies (Hern 1982; Hogue, Cates, and Tietze 1982). An uncomplicated early abortion should have no effect on future health or childbearing. If the abortion permits postponement of the first term pregnancy to after adolescence, the usual risks associated with a Strong emotional support and a knowledgable physician can ease the difficulties women face when dealing with abortion. JENNIE WOODCOCK; REFLECTIONS PHOTOLIBRARY/CORBIS first term pregnancy are actually reduced. Psychological studies consistently show that women who are basically healthy can adjust to any outcome of pregnancy, whether it is term birth, induced abortion, or spontaneous abortion (miscarriage) (Adler et al. 1990). It is highly desirable, however, to have strong emotional support not only from friends and family but also from a sympathetic physician and a lay abortion counselor who will be with the woman during her abortion experience.

Denial of abortion can have serious adverse consequences for the children who result from the pregnancies their mothers had wanted to terminate. A long-term study in Czechoslovakia of the offspring of women who were denied abortions showed a range of adjustment and developmental difficulties in these children (David et al. 1988).

Incedence of abortion

Although the exact number may never be known, it is estimated that between 20 million and 50 million abortions are performed each year (World Health Organization 1994). The proportion of women having abortions and the proportion of pregnancies terminated vary widely from country to country. In the past, the highest rates have been observed in the Soviet Union and eastern European countries where abortion is more socially acceptable than in other regions and where contraceptive services have been scarce or unreliable.

According to Singh and Henshaw (1996), about half of all abortions in 1990 occurred in Asia, with almost one-fourth occurring in the former USSR. Approximately 3 percent occurred in Canada and the United States. In Colombia during the 1980s, according to unofficial reports, it appeared that one out of every two pregnancies ended in abortion.

The highest abortion rates recorded have been in Romania in 1965, where, among women in the reproductive age from fifteen to forty-four, one in four had an abortion each year (Henshaw and Morrow 1990). The abortion rate in Romania plummeted in 1966 when Romanian dictator Nicolau Ceaucescu banned abortion in an attempt to increase population growth rates. Police surveillance of women included mandatory pelvic examinations and pregnancy tests. This action resulted in higher birth rates, but it was also accompanied by skyrocketing maternal mortality rates including a dramatic increase in deaths from abortion, which caused approximately 85 percent of all maternal deaths. The Romanian maternal mortality rate went from 86 per 100,000 live births in 1966 to 170 per 100,000 live births in the late 1980s—the highest in Europe.

Approximately 10,000 excess maternal deaths due to abortion occurred during the period from 1966–1989 (Serbanescu et al. 2001). Romanian abortion rates again became the highest in the world after Ceaucescu was overthrown in 1989, and abortion mortality rates dropped ( Joffe 1999; Henshaw 1999). Within one year after the fall of the Ceaucescu regime, the maternal mortality rate dropped by 50 per cent. By 1997, there were 21 abortion-related deaths per 100,000 live births (Serbanescu et al. 2001).

In other countries such as Canada and the Netherlands, where abortion is legal and widely available, but where other means of fertility control are easily available, abortion rates are sometimes quite low (Henshaw 1999). In the Ukraine, the abortion rate in women in the reproductive age range of fifteen to forty-four years fell 50 percent from 77 abortions per 1,000 women to 36 per 1,000 in the interval from 1990 to 1998 (Goldberg et al. 2001).

It appears that, when abortion is both legal and widely available but is not the only means of effective fertility control, about one-fourth of all pregnancies will end in abortion. Lack of access to contraception may result in higher abortion rates. The principal effect of laws making abortion illegal appears to be to make abortion more dangerous but not less common.

Risk of Abortion

In the United States, Canada, and Western Europe, abortion has become not only the most common but also one of the safest operations being performed. This was not always the case. In the nineteenth and early twentieth centuries, abortion was quite dangerous, and many women died as a result.

Pregnancy itself is not a harmless condition; women can die during pregnancy. The maternal mortality rate (the proportion of women dying from pregnancy and childbirth) is found by dividing the number of women dying from all causes related to pregnancy, childbirth, and the puerperium (the six-week period following childbirth) by the total number of live births and then multiplying by a constant factor such as 100,000. For example, the maternal mortality rate in the United States in 1920 was 680 maternal deaths per 100,000 live births (Lerner and Anderson 1963). It had fallen to 38 deaths per 100,000 live births by 1960 and 8 deaths per 100,000 live births by 1994. Illegal abortion accounted for about 50 percent of all maternal deaths in 1920, and that was still true in 1960. By 1980, however, the percentage of deaths due to abortion had dropped to nearly zero (Cates 1982). The difference in maternal mortality rates due to abortion reflected the increasing legalization of abortion from 1967 to 1973 that permitted abortions to be done safely by doctors in clinics and hospitals. The changed legal climate also permitted the prompt treatment of complications that occurred with abortions.

The complication rates and death rates associated with abortion itself can also be examined. In 1970, Christopher Tietze of the Population Council began studying the risks of death and complications due to abortion by collecting data from hospitals and clinics throughout the nation. The statistical analyses at that time showed that the death rate due to abortion was about 2 deaths per 100,000 procedures compared with the current maternal mortality rate exclusive of abortion of 12 deaths per 100,000 live births. In other words, a woman having an abortion was six times less likely to die than a woman who chose to carry a pregnancy to term. Tietze also found that early abortion was many times safer than abortion done after twelve weeks of pregnancy (Tietze and Lewit 1972) and that some abortion techniques were safer than others. The Centers for Disease Control in Atlanta took over the national study of abortion statistics that had been developed by Tietze, and abortion became the most carefully studied surgical procedure in the United States. As doctors gained more experience with abortion and as techniques improved, death and complication rates due to abortion continued to decline. The rates declined because women were seeking abortions earlier in pregnancy, when the procedure was safer. Clinics where safe abortions could be obtained were opened in many U.S. cities across the country, improving access to this service.

By the early 1990s in the United States, the risk of death in early abortion was less than 1 death per 1 million procedures, and for later abortion, about 1 death per 100,000 procedures (Koonin et al. 1992). The overall risk of death in abortion was about 0.4 deaths per 100,000 procedures compared with a maternal mortality rate (exclusive of abortion) of about 9.1 deaths per 100,000 live births (Koonin et al. 1991a, 1991b).

When and how abortion are performed

In the United States and in European countries such as the Netherlands, more than 90 percent of all abortions are performed in the first trimester of pregnancy (up to twelve weeks from the last normal menstrual period). Most take place in outpatient clinics specially designed and equipped for this purpose. Nearly all abortions in the United States are performed by physicians, although two states (Montana and Vermont) permit physicians' assistants to do the procedure. A limited number of physicians in specialized clinics perform abortions during the second trimester of pregnancy, but only a few perform abortions after pregnancy has advanced to more than twenty-five weeks. Although hospitals permit abortions to be performed, the number is limited because the costs to perform an abortion in the hospital are greater and hospital operating room schedules do not allow for a large number of patients. In addition, staff members at hospitals are not chosen on the basis of their willingness to help perform abortions, while clinic staff members are hired for that purpose.

Most early abortions (up to twelve to fourteen weeks of pregnancy) are performed with some use of vacuum aspiration equipment. A machine or specially designed syringe is used to create a vacuum, and the suction draws the contents of the uterus into an outside container. The physician then checks the inside of the uterus with a curette, a spoon-shaped device with a loop at the end and sharp edges to scrape the wall of the uterus (Hern 1990).

Before the uterus can be emptied, however, the cervix (opening of the uterus) must be dilated, or stretched, in order to introduce the instruments. There are two principal ways this can be done. Specially designed metal dilators, steel rods with tapered ends that allow the surgeon to force the cervix open a little at a time, are used for most abortions. This process is usually done under local anesthesia, but sometimes general anesthesia is used. The cervix can also be dilated by placing pieces of medically prepared seaweed stalk called Laminaria in the cervix and leaving it for a few hours or overnight (Hern 1975, 1990). The Laminaria draws water from the woman's tissues and swells up, gently expanding as the woman's cervix softens and opens from the loss of moisture. The Laminaria is then removed, and a vacuum cannula or tube is placed into the uterus to remove the pregnancy by suction (Figure 1). Following this, the walls of the uterus are gently scraped with the curette.

After twelve weeks of pregnancy, performing an abortion becomes much more complicated and dangerous. The uterus, the embryo or fetus, and the blood vessels within the uterus are all much larger. The volume of amniotic fluid around the fetus has increased substantially, creating a potential hazard. If the amniotic fluid enters the woman's circulatory system, she could die instantly or bleed to death from a disruption of the blood-clotting system. This hazard is an important consideration in performing late abortions.



Ultrasound equipment, which uses sound waves to show a picture of the fetus, is used to examine the woman before a late abortion is performed. Parts of the fetus such as the head and long bones are measured to determine the length of pregnancy. The ultrasound image also permits determination of fetal position, location of the placenta, and the presence of any abnormalities that could cause a complication.

Between fourteen and twenty weeks of pregnancy, Laminaria is placed in the cervix over a period of a day or two, sometimes changing the Laminaria and replacing the first batch with a larger amount in order to increase cervical dilation (Hern 1990). At the time of the abortion, the Laminaria is removed, the amniotic sac (bag of waters) is ruptured with an instrument, and the amniotic fluid is allowed to drain out. This procedure reduces the risk of an amniotic fluid embolism, escape of the amniotic fluid into the bloodstream, and allows the uterus to contract to make the abortion safer. Using an ultrasound real-time image, the surgeon then places special instruments such as grasping forceps into the uterus and removes the fetus and placenta (Hern 1990). This has proven to be the safest way to perform late abortions, but it requires great care and skill.

Other methods of late abortion include the use of prostaglandin (a naturally occurring hormone), either by suppository or by injection (Hern 1988). Other materials injected into the pregnant uterus to effect late abortion include hypertonic (concentrated) saline (salt) solution, hypertonic urea, and hyperosmolar (concentrated) glucose solution.

Injections are also used with late abortions, especially those performed at twenty-five weeks or more for reasons of fetal disorder. The lethal injection into the fetus is performed several days prior to the abortion, along with other treatments that permit a safe abortion (Hern et al. 1993; Hern 2001).

Although surgical abortion is still performed outside the United States, medical abortion is growing in use in Europe and in the United States following the introduction in France in 1988 of mifepristone (also known as RU-486) and misoprostol, a synthetic prostaglandin. Mifepristone works by blocking the hormonal receptors in the placenta from receiving progesterone, which is necessary for continuation of the pregnancy. Along with misoprostol, mifepristone may cause a complete abortion in 95 percent of early pregnancies within a few days. Most patients do not require a surgical treatment for completion of the abortion.

Reasons for Abortion

There are probably as many reasons for abortions as there are women who have them. Some pregnancies result from rape or incest, and women who are victims of these assaults often seek abortions. Most women, however, decide to have an abortion because the pregnancy represents a problem in their lives (Bankole et al. 1998, 1999; Alan Guttmacher Institute 1999).

Some women feel emotionally unprepared to enter parenthood and raise a child; they are too young or do not have a reliable partner with whom to raise a child. Many young women in high school or college find themselves pregnant and must choose between continuing the education they need to survive economically and dropping out to have a baby. Young couples who are just starting their lives together and want children might prefer to become financially secure first to provide better care for their future children.

Sometimes people enter into a casual sexual relationship that leads to pregnancy with no prospect of marriage. Even if the sexual relationship is more than casual, abortion is may be sought because a woman decides that the social status of the male is inappropriate. Abortion is reported to be sought by some women because of popular beliefs that forms of modern contraceptives are more dangerous than abortion (Otoide et al. 2001).

Some of the most difficult and painful choices are faced by women who are happily pregnant for the first time late in the reproductive years (thirty-five to forty-five) but discover in late pregnancy (twenty-six or more weeks) that the fetus is so defective it may not live or have a normal life. Even worse is a diagnosis of abnormalities that may or may not result in problems after birth. Some women and couples in this situation choose to have a late abortion (Kolata 1992; Hern et al. 1993).

In some cases, a woman must have an abortion to survive a pregnancy. An example is the diabetic woman who develops a condition in pregnancy called hyperemesis gravidarum (uncontrollable vomiting associated with pregnancy). She becomes malnourished and dehydrated in spite of intravenous therapy and other treatment, threatening heart failure, among other things. Only an abortion will cure this life-threatening condition.

In certain traditional or tribal societies, either the decision to end a pregnancy by abortion or the method of doing so is determined by the group. John Early and John Peters (1990) described a method used by the Yanomami of the Amazon of hitting or jumping on a pregnant woman's abdomen to cause an abortion. A similar method has been described in other tribal societies in Africa and South Asia. Among the Suraya of seventeenth-century Taiwan, a woman under the age of thirty was required to end all pregnancies by abortion by forceful uterine massage (Shepherd 1995).

Studies done in Chile in the 1960s showed that the majority of women who sought

Definition of Abortion

Abortion is one of the most difficult, controversial, and painful subjects in modern society. The principal controversy revolves around the questions of who makes the decision concerning abortion, the individual or the state; under what circumstances it may be done; and who is capable of making the decision. Medical questions such as techniques of abortion are less controversial but are sometimes part of the larger debate.

Abortion is not new in human society; a study by the anthropologist George Devereux (1955) showed that more than 300 contemporary human nonindustrial societies practiced abortion. Women have performed abortions on themselves or experienced abortions at the hands of others for thousands of years (Potts, Diggory, and Peel 1977), and abortions continue to occur today in developing areas under medically primitive conditions. However, modern technology and social change have made abortion a part of modern health care. At the same time, abortion has become a political issue in some societies and a flash point for disagreements about the role of women and individual autonomy in life decisions.



The various social responses to abortion range from those of the individual and her immediate circle of family and friends to the organizational, community, and even national levels. Each culture and society has specific ways of dealing with unplanned or unwanted pregnancy and with abortion. These traditions are changing rapidly in the modern world.

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TO TEACH YOUR STUDENTS HOW TO SOLVE PROBLEMS
1. Give your students problems that they CAN solve
This is basic! The joy of problem solving is SOLVING problems. The misery of problem solving is failing to solve problems. Students who experience the JOY of solving problems WANT to solve MORE problems. Students who experience only the misery of failure DO NOT even want to TRY to solve any more problems. To be good problem solvers, students must BELIEVE that they can solve problems. Therefore you must give them problems they CAN SOLVE.
2. Give your students problems that SEEM difficult.
This also is basic! If ALL the problems that your students solve are EASY problems, they will not believe they can solve difficult problems. Therefore you must TEACH them to solve problems that SEEM difficult but CAN BE SOLVED--if they try! Problems only SEEM difficult before they’ve been solved. After they’ve been solved, problems seem easy.
3. Teach your students HOW to solve problems that SEEM difficult.
Problems seem difficult when the solution is not obvious. Usually, difficult problems are solved by a series of logical STEPS that CANNOT BE SEEN—at first! If all the steps of the solution can be seen immediately, the problem is not difficult. But good problem solvers START by TRYING to solve the problem. They take the FIRST STEP, and look for the second step. If they do not see the second step, they take a DIFFERENT first step. In other words good problem solvers start by TRYING to solve problems that seem difficult, not by wondering how to solve them.
4. Teach your students problem-solving STRATEGIES.
The choice of a problem solving STRATEGY suggests the first step and helps the problem solver to see the next step. Good problem solvers do not give up; they try a different strategy, a different first step, and then they look for the second step.
PROBLEM-SOLVING STRATEGIES DISCUSSED IN THIS WEBSITE
Draw a Diagram
Make a List
Guess and Check
Divide and Conquer
Look for a Pattern
Start at the End
5. At first, tell them which strategy to use, but later, expect them to CHOOSE AND TRY to find an effective strategy for themselves.
Introduce your students to each of the problem-solving strategies. You will find sample problems for each strategy and sample solutions showing you how experienced students have solved them. At first, tell your students which strategy to use. In this way, the problems are made a little easier, and they will get some practice using each strategy in turn. Eventually, you will not tell them which strategy to use, and for this purpose you may select from the “Mixed Problems.”

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Math Problem Solving

Math problem solving is a cognitive mental activity that involves several complex strategies and processes. The systems of solving mathematical problems consist of two stages: representation of problems and execution of problems. Math problem recognition is important to make out and work out problems successfully. Students who find it difficult to recognize mathematical problems may face difficulty while solving them. They should keep strategies for recognizing the particular type of problem at their fingers’ end.

The Use of Visualization in Solving Mathematical Problems

An effective and efficient strategy to illustrate math problems is visualization. Visual representation of a mathematical problem in one’s mind or on a piece of paper shows the links among several segments of the problem. It is called schematic representation. Students who are poor in solving math problems make immature and improper representations that are more pictorial and less schematic.

The use of visual representation in solving math problems is beyond the knowledge of many students. Some students apply the power of visualization inappropriately and ineffectively. It is also true about many teachers. They rely on mere math problem solving guidebooks to teach students how to solve mathematical problems.

Strategies Of Solving Mathematical ProblemsReading, understanding and paraphrasing mathematical problems for successful math problem solving are other cognitive strategies. Students should be taught to hypothesize a plan, assess the process and estimate an outcome of their attempts to solve problems. The process in which math problem solving is done involves not only cognitive strategies such as estimation and visualization but some self-regulation strategies as well.

Good math problems solvers go for different strategies to work out problems. They read to understand and represent to solve them. On the basis of their understanding they translate the numerical and linguistical information in the problems into mathematical notations. They go through the problems again and again to get to the bottom of them. Then they meditate on the process of solving them.

Students with inefficiency in solving mathematical problems do not find the underlying cause of problem. They lack the resources, which are essential for execution of this complex cerebral activity. Attempts made by such students generally lack self-regulation process or metacognitive. So their endeavors to comprehend, analyze and solve problems go into water. To learn and teach cognitive strategies and metacognitive processes is the maker of a good problem solver.

Math problem solving is both a good mental exercise as well as brain booster. It makes the mind inventive and the brain fertile. Practicing it regularly nourishes students’ logical bent of mind. The curriculum of math problem solving is one of the most essentials part of education.

MATHEMATICAL PROBLEM SOLVING

byJames W. Wilson, Maria L. 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. (26, p. v.)

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 problem 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.
How long is the groove on one side of a long-play (33 1/3 rpm) phonograph record? Assume there is a single recording and the Outer (beginning) groove is 5.75 inches from the center and the Inner (ending) groove is 1.75 inches from the center. 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. 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.e., a problem approach. 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. 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. Various 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."
A nine-digit number is formed using each of the digits 1,2,3,...,9 exactly once. For n = 1,2,3,...,9, n divides the first n digits of the number. Find the number.
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 (26,27,28), the 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 a 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 (25), Larkin (18), and 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. Most formulations of a problem solving framework in U. S. textbooks attribute some relationship to Polya's (26) problem solving stages. 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 (28), problem solving was a major theme of doing mathematics and "teaching students to think" was of primary importance. "How to think" is a theme that underlies much of genuine inquiry and problem solving in mathematics. 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." This is, in particular, a byproduct of an emphasis on procedural knowledge about problem solving as seen in the linear frameworks of U. S. mathematics textbooks (Figure 1) and the very limited problems/exercises included in lessons. Figure 1: Linear models of problem solving found in textbooks that are inconsistent with genuine problem solving.
Figure 1
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:
1. They depict problem solving as a linear process.2. They present problem solving as a series of steps.3. They imply that solving mathematics problems is a procedure to be memorized, practiced, and habituated.4. 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 solving 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. Another 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. S. mathematics education in recent years. 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
Figure 2
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. 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."
Algorithms
An algorithm is a procedure, applicable to a particular type of exercise, which, if followed correctly, is guaranteed to give you the answer to the exercise. 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
The creation of an algorithm may involve developing a process for factoring quadratic equations, as well as developing a process for partitioning a line segment using only Euclidian constructions.
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. Research involving this approach is currently more prevalent at the elementary level within the context of constructivist theories.
Heuristics
Heuristics are kinds of information, available to students in making decisions during problem solving, that are aids to the generation of a solution, plausible in nature rather than prescriptive, seldom providing infallible guidance, and variable in results. Somewhat synonymous terms are strategies, techniques, and rules-of-thumb. For example, admonitions to "simplify an algebraic expression by removing parentheses," to "make a table," to "restate the problem in your own words," or to "draw a figure to suggest the line of argument for a proof" are heuristic in nature. Out of context, they have no particular value, but incorporated into situations of doing mathematics they can be quite powerful (26,27,28).
Many a guess has turned out to be wrong but nevertheless useful in leading to a better one. Polya (26, p. 99)
Theories of mathematics problem solving (25,33,50) have placed a major focus on the role of heuristics. Surely it seems that providing explicit instruction on the development and use of heuristics should enhance problem solving performance; yet it is not that simple. Schoenfeld (35) and Lesh (19) have pointed out the limitations of such a simplistic analysis. Theories must be enlarged to incorporate classroom contexts, past knowledge and experience, and beliefs. What Polya (26) describes in How to Solve It is far more complex than any theories we have developed so far. Mathematics instruction stressing heuristic processes has been the focus of several studies. Kantowski (14) used heuristic instruction to enhance the geometry problem solving performance of secondary school students. Wilson (50) and Smith (42) examined contrasts of general and task specific heuristics. These studies revealed that task specific hueristic instruction was more effective than general hueristic instruction. Jensen (12) used the heuristic of subgoal generation to enable students to form problem solving plans. He used thinking aloud, peer interaction, playing the role of teacher, and direct instruction to develop students' abilities to generate subgoals.
Managing It All
An extensive knowledge base of domain specific information, algorithms, and a repertoire of heuristics are not sufficient during problem solving. The student must also construct some decision mechanism to select from among the available heuristics, or to develop new ones, as problem situations are encountered. A major theme of Polya's writing was to do mathematics, to reflect on problems solved or attempted, and to think (27,28). Certainly Polya expected students to engage in thinking about the various tactics, patterns, techniques, and strategies available to them. To build a theory of problem solving that approaches Polya's model, a manager function must be incorporated into the system. Long ago, Dewey (8), in How we think, emphasized self-reflection in the solving of problems. Recent research has been much more explicit in attending to this aspect of problem solving and the learning of mathematics. The field of metacognition concerns thinking about one's own cognition. Metacognition theory holds that such thought can monitor, direct, and control one's cognitive processes (4,41). Schoenfeld (34) described and demonstrated an executive or monitor component to his problem solving theory. His problem solving courses included explicit attention to a set of guidelines for reflecting about the problem solving activities in which the students were engaged. Clearly, effective problem solving instruction must provide the students with an opportunity to reflect during problem solving activities in a systematic and constructive way.
The Importance of Looking Back
Looking back may be the most important part of problem solving. It is the set of activities that provides the primary opportunity for students to learn from the problem. The phase was identified by Polya (26) with admonitions to examine the solution by such activities as checking the result, checking the argument, deriving the result differently, using the result, or the method, for some other problem, reinterpreting the problem, interpreting the result, or stating a new problem to solve. Teachers and researchers report, however, that developing the disposition to look back is very hard to accomplish with students. Kantowski (14) found little evidence among students of looking back even though the instruction had stressed it. Wilson (51) conducted a year long inservice mathematics problem solving course for secondary teachers in which each participant developed materials to implement some aspect of problem solving in their on-going teaching assignment. During the debriefing session at the final meeting, a teacher put it succinctly: "In schools, there is no looking back." The discussion underscored the agreement of all the participants that getting students to engage in looking back activities was difficult. Some of the reasons cited were entrenched beliefs that problem solving in mathematics is answer getting; pressure to cover a prescribed course syllabus; testing (or the absence of tests that measure processes); and student frustration. The importance of looking back, however, outweighs these difficulties. Five activities essential to promote learning from problem solving are developing and exploring problem contexts, extending problems, extending solutions, extending processes, and developing self-reflection. Teachers can easily incorporate the use of writing in mathematics into the looking back phase of problem solving. It is what you learn after you have solved the problem that really counts.
Problem Posing
Problem posing (3) and problem formulation (16) are logically and philosophically appealing notions to mathematics educators and teachers. Brown and Walter provide suggestions for implementing these ideas. In particular, they discuss the "What-If-Not" problem posing strategy that encourages the generation of new problems by changing the conditions of a current problem. For example, given a mathematics theorem or rule, students may be asked to list its attributes. After a discussion of the attributes, the teacher may ask "what if some or all of the given attributes are not true?" Through this discussion, the students generate new problems. Brown and Walter provide a wide variety of situations implementing this strategy including a discussion of the development of non-Euclidean geometry. After many years of attempting to prove the parallel postulate as a theorem, mathematicians began to ask "What if it were not the case that through a given external point there was exactly one line parallel to the given line? What if there were two? None? What would that do to the structure of geometry?" (p.47). Although these ideas seem promising, there is little explicit research reported on problem posing.
Brown and Walter. (3)
Problem Solving as an Instructional Goal
What is mathematics?
If our answer to this question uses words like exploration, inquiry, discovery, plausible reasoning, or problem solving, then we are attending to the processes of mathematics. Most of us would also make a content list like algebra, geometry, number, probability, statistics, or calculus. Deep down, our answers to questions such as What is mathematics? What do mathematicians do? What do mathematics students do? Should the activities for mathematics students model what mathematicians do? can affect how we approach mathematics problems and how we teach mathematics. The National Council of Teachers of Mathematics (NCTM) (23,24) recommendations to make problem solving the focus of school mathematics posed fundamental questions about the nature of school mathematics. The art of problem solving is the heart of mathematics. Thus, mathematics instruction should be designed so that students experience mathematics as problem solving.
The National Council of Teachers of Mathematics recommends that --l. problem solving be the focus of school mathematics in the 1980s. An Agenda for Action (23) We strongly endorse the first recommendation of An Agenda for Action. The initial standard of each of the three levels addresses this goal. Curriculum and Evaluation Standards (24)
Why Problem Solving?
The NCTM (23,24) has strongly endorsed the inclusion of problem solving in school mathematics. There are many reasons for doing this.First, problem solving is a major part of mathematics. It is the sum and substance of our discipline and to reduce the discipline to a set of exercises and skills devoid of problem solving is misrepresenting mathematics as a discipline and shortchanging the students. Second, mathematics has many applications and often those applications represent important problems in mathematics. Our subject is used in the work, understanding, and communication within other disciplines. Third, there is an intrinsic motivation embedded in solving mathematics problems. We include problem solving in school mathematics because it can stimulate the interest and enthusiasm of the students. Fourth, problem solving can be fun. Many of us do mathematics problems for recreation. Finally, problem solving must be in the school mathematics curriculum to allow students to develop the art of problem solving. This art is so essential to understanding mathematics and appreciating mathematics that it must be an instructional goal. Teachers often provide strong rationale for not including problem solving activities is school mathematics instruction. These include arguments that problem solving is too difficult, problem solving takes too much time, the school curriculum is very full and there is no room for problem solving, problem solving will not be measured and tested, mathematics is sequential and students must master facts, procedures, and algorithms, appropriate mathematics problems are not available, problem solving is not in the textbooks, and basic facts must be mastered through drill and practice before attempting the use of problem solving. We should note, however, that the student benefits from incorporating problem solving into the mathematics curriculum as discussed above outweigh this line of reasoning. Also we should caution against claiming an emphasize on problem solving when in fact the emphasis is on routine exercises. From various studies involving problem solving instruction, Suydam (44) concluded:
If problem solving is treated as "apply the procedure," then the students try to follow the rules in subsequent problems. If you teach problem solving as an approach, where you must think and can apply anything that works, then students are likely to be less rigid. (p. 104)
Problem Solving as an Instructional Method
Problem solving as a method of teaching may be used to accomplish the instructional goals of learning basic facts, concepts, and procedures, as well as goals for problem solving within problem contexts. For example, if students investigate the areas of all triangles having a fixed perimeter of 60 units, the problem solving activities should provide ample practice in computational skills and use of formulas and procedures, as well as opportunities for the conceptual development of the relationships between area and perimeter. The "problem" might be to find the triangle with the most area, the areas of triangles with integer sides, or a triangle with area numerically equal to the perimeter. Thus problem solving as a method of teaching can be used to introduce concepts through lessons involving exploration and discovery. The creation of an algorithm, and its refinement, is also a complex problem solving task which can be accomplished through the problem approach to teaching. Open ended problem solving often uses problem contexts, where a sequence of related problems might be explored. For example, the problems in the margins evolved from considering gardens of different shapes that could be enclosed with 100 yards of fencing.
Suppose one had 100 yards of fencing to enclose a garden. What shapes could be enclosed? What are the dimensions of each and what is the area? Make a chart.What triangular region with P = 100 has the most area?Find all five triangular regions with P = 100 having integer sides and integer area. (such as 29, 29, 42)What rectangular regions could be enclosed? Areas? Organize a table? Make a graph?Which rectangular region has the most area? from a table? from a graph? from algebra, using the arithmetic mean-geometric mean inequality?What is the area of a regular hexagon with P = 100?What is the area of a regular octagon with P = 100?What is the area of a regular n-gon with P = 100? Make a table for n = 3 to 25. Make a graph. What happens to 1/n(tan 180/n) as n increases?What if part of the fencing is used to build a partition perpendicular to a side? Consider a rectangular region with one partition? With 2 partitions? with n partitions? (There is a surprise in this one!!) What if the partition is a diagonal of the rectangle?What is the maximum area of a sector of a circle with P = 100? (Here is another surprise!!! -- could you believe it is r2 when r = 25? How is this similar to a square being the maximum rectangle and the central angle of the maximum sector being 2 radians?)What about regions built along a natural boundary? For example the maximum for both a rectangular region and a triangular region built along a natural boundary with 100 yards of fencing is 1250 sq. yds. But the rectangle is not the maximum area four-sided figure that can be built. What is the maximum-area four-sided figure?
Many teachers in our workshops have reported success with a "problem of the week" strategy. This is often associated with a bulletin board in which a challenge problem is presented on a regular basis (e.g., every Monday). The idea is to capitalize on intrinsic motivation and accomplishment, to use competition in a constructive way, and to extend the curriculum. Some teachers have used schemes for granting "extra credit" to successful students. The monthly calendar found in each issue of The Mathematics Teacher is an excellent source of problems. Whether the students encounter good mathematics problems depends on the skill of the teacher to incorporate problems from various sources (often not in textbooks). We encourage teachers to begin building a resource book of problems oriented specifically to a course in their on-going workload. Good problems can be found in the Applications in Mathematics (AIM Project) materials (21) consisting of video tapes, resource books and computer diskettes published by the Mathematical Association of America. These problems can often be extended or modified by teachers and students to emphasize their interests. Problems of interest for teachers and their students can also be developed through the use of The Challenge of the Unknown materials (1) developed by the American Association for the Advancement of Science. These materials consist of tapes providing real situations from which mathematical problems arise and a handbook of ideas and activities that can be used to generate other problems.
Beliefs about Mathematics Problem Solving
The importance of students' (and teachers') beliefs about mathematics problem solving lies in the assumption of some connection between beliefs and behavior. Thus, it is argued, the beliefs of mathematics students, mathematics teachers, parents, policy makers, and the general public about the roles of problem solving in mathematics become prerequisite or co-requisite to developing problem solving. The Curriculum and Evaluation Standards makes the point that "students need to view themselves as capable of using their growing mathematical knowledge to make sense of new problem situations in the world around them" (24, p. ix.). We prefer to think of developing a sense of "can do" in our students as they encounter mathematics problems.
The first rule of teaching is to know what you are supposed to teach. The second rule of teaching is to know a little more than what you are supposed to teach. . . . Yet it should not be forgotten that a teacher of mathematics should know some mathematics, and that a teacher wishing to impart the right attitude of mind toward problems to his students should have acquired that attitude himself. Polya (26, p. 173).
Schoenfeld (36,37) reported results from a year-long study of detailed observations, analysis of videotaped instruction, and follow-up questionnaire data from two tenth-grade geometry classes. These classes were in select high schools and the classes were highly successful as determined by student performance on the New York State Regent's examination. Students reported beliefs that mathematics helps them to think clearly and they can be creative in mathematics, yet, they also claimed that mathematics is learned best by memorization. Similar contrasts have been reported for the National Assessment (5). Indeed our conversations with teachers and our observations portray an overwhelming predisposition of secondary school mathematics students to view problem solving as answer getting, view mathematics as a set of rules, and be highly oriented to doing well on tests. Schoenfeld (37) was able to tell us much more about the classes in his study. He makes the following points.
The rhetoric of problem solving has become familiar over the past decade. That rhetoric was frequently heard in the classes we observed -- but the reality of those classrooms is that real problems were few and far between . . . virtually all problems the students were asked to solve were bite-size exercises designed to achieve subject matter mastery: the exceptions were clearly peripheral tasks that the students found enjoyable but that they considered to be recreations or rewards rather than the substance they were expected to learn . . . the advances in mathematics education in the [past] decade . . . have been largely in our acquiring a more enlightened goal structure, and having students pick up the rhetoric -- but not the substance -- related to those goals. (pp. 359-9)
Each of us needs to ask if the situation Schoenfeld describes is similar to our own school. We must take care that espoused beliefs about problem solving are consistent with a legitimately implemented problem solving focus in school mathematics.
Technology and Problem Solving
The appropriate use of technology for many people has significant identity with mathematics problem solving. This view emphasizes the importance of technology as a tool for mathematics problem solving. This is in contrast to uses of technology to deliver instruction or for generating student feedback.
Programming as Problem Solving
In the past, problem solving research involving technology has often dealt with programming as a major focus. This research has often provided inconclusive results. Indeed, the development of a computer program to perform a mathematical task can be a challenging mathematical problem and can enhance the programmer's understanding of the mathematics being used. Too often, however, the focus is on programming skills rather than on using programming to solve mathematics problems. There is a place for programming within mathematics study, but the focus ought to be on the mathematics problems and the use of the computer as a tool for mathematics problem solving.
A ladder 5 meters long leans against a wall, reaching over the top of a box that is 1 meter on each side. The box is against the wall. What is the maximum height on the wall that the ladder can reach? The side view is:Assume the wall is perpendicular to the floor. Use your calculator to find the maximum height to the nearest .01 meter.
Iteration
Iteration and recursion are concepts of mathematics made available to the secondary school level by technology. Students may implement iteration by writing a computer program, developing a procedure for using a calculator, writing a sequence of decision steps, or developing a classroom dramatization. The approximation of roots of equations can be made operational with a calculator or computer to carry out the iteration. For example, the process for finding the three roots of

is not very approachable without iterative techniques. Iteration is also useful when determining the maximum height, h, between a chord and an arc of a circle when the length S of the arc and the length L of the chord are known. This may call for solving

simultaneously and using iterative techniques to find the radius r and and central angle ø in order to evaluate h = r - r cos ø. Fractals can also be explored through the use of iterative techniques and computer software.
Exploration
Technology can be used to enhance or make possible exploration of conceptual or problem situations. For example, a function grapher computer program or a graphics calculator can allow student exploration of families of curves such as
for different values of a, b, and c. A calculator can be used to explore sequences such as
for different values of a. In this way, technology introduces a dynamic aspect to investigating mathematics. Thomas (46) studied the use of computer graphic problem solving activities to assist in the instruction of functions and transformational geometry at the secondary school level. The students were challenged to create a computer graphics design of a preselected picture using graphs of functions and transformational geometry. Thomas found these activities helped students to better understand function concepts and improved student attitudes.
Evaluation of Problem Solving
As the emphasis on problem solving in mathematics classrooms increases, the need for evaluation of progress and instruction in problem solving becomes more pressing. It no longer suffices for us to know which kinds of problems are correctly and incorrectly solved by students. As Schoenfeld (36) describes:
All too often we focus on a narrow collection of well-defined tasks and train students to execute those tasks in a routine, if not algorithmic fashion. Then we test the students on tasks that are very close to the ones they have been taught. If they succeed on those problems, we and they congratulate each other on the fact that they have learned some powerful mathematical techniques. In fact, they may be able to use such techniques mechanically while lacking some rudimentary thinking skills. To allow them, and ourselves, to believe that they "understand" the mathematics is deceptive and fraudulent. (p. 30)
Schoenfeld (31) indicates that capable mathematics students when removed from the context of coursework have difficulty doing what may be considered elementary mathematics for their level of achievement. For example, he describes a situation in which he gave a straightforward theorem from tenth grade plane geometry to a group of junior and senior mathematics majors at the University of California involved in a problem solving course. Of the eight students solving this problem only two made any significant progress. We need to focus on the teaching and learning of mathematics and, in turn, problem solving using a holistic approach. As recommended in the NCTM's An Agenda for Action (23), "the success of mathematics programs and student learning [must] be evaluated by a wider range of measures than conventional testing" (p. 1). Although this recommendation is widely accepted among mathematics educators, there is a limited amount of research dealing with the evaluation of problem solving within the classroom environment.
Classroom research: Ask your students to keep a problem solving notebook in which they record on a weekly basis:
(1) their solution to a mathematics problem.(2) a discussion of the strategies they used to solve the problem.(3) a discussion of the mathematical similarities of this problem with other problems they have solved.(4) a discussion of possible extensions for the problem.(5) an investigation of at least one of the extensions they discussed.
Use these notebooks to evaluate students' progress. Then periodically throughout the year, analyze the students' overall progress as well as their reactions to the notebooks in order to asses the effectiveness of the evaluation process.
Some research dealing with the evaluation of problem solving involves diagnosing students' cognitive processes by evaluating the amount and type of help needed by an individual during a problem solving activity. Campione, Brown, and Connell (4) term this method of evaluation as dynamic assessment. Students are given mathematics problems to solve. The assessor then begins to provide as little help as necessary to the students throughout their problem solving activity. The amount and type of help needed can provide good insight into the students' problem solving abilities, as well as their ability to learn and apply new principles. Trismen (47) reported the use of hints to diagnosis student difficulties in problem solving in high school algebra and plane geometry. Problems were developed such that the methods of solutions where not readily apparent to the students. A sequence of hints was then developed for each item. According to Trismen, "the power of the hint technique seems to lie in its ability to identify those particular students in need of special kinds of help" (p. 371). Campione and his colleagues (4) also discussed a method to help monitor and evaluate the progress of a small cooperative group during a problem solving session. A learning leader (sometimes the teacher sometimes a student) guides the group in solving the problem through the use of three boards: (1) a Planning Board, where important information and ideas about the problem are recorded, (2) a Representation Board, where diagrams illustrating the problems are drawn, and (3) a Doing Board, where appropriate equations are developed and the problem is solved. Through the use of this method, the students are able to discuss and reflect on their approaches by visually tracing their joint work. Campione and his colleagues indicated that increased student engagement and enthusiasm in problem solving, as well as, increased performance resulted from the use of this method for solving problems. Methods, such as the clinical approach discussed earlier, used to gather data dealing with problem solving and individual's thinking processes may also be used in the classroom to evaluate progress in problem solving. Charles, Lester, and O'Daffer (7) describe how we may incorporate these techniques into a classroom problem solving evaluation program. For example, thinking aloud may be canonically achieved within the classroom by placing the students in cooperative groups. In this way, students may express their problem solving strategies aloud and thus we may be able to assess their thinking processes and attitudes unobtrusively. Charles and his colleagues also discussed the use of interviews and student self reports during which students are asked to reflect on their problem solving experience a technique often used in problem solving research. Other techniques which they describe involve methods of scoring students' written work. Figure 3 illustrates a final assignment used to assess teachers' learning in a problem solving course that has been modified to be used with students at the secondary level.
1. (20 points)Select a problem that you have worked on but not yet solved, and that you feel you can eventually solve. Present the following:
a. Show or describe what you have done so far (It could be that you tell me where to find your work in your notebook).b. Assess how you feel about the problem. Is the lack of closure a concern? Why?c. Assess what you may have learned in working on the problem so far.
2. (20 points)
a. Select a mathematics theorem or rule from class and make a list of its attributes?b. Generate at least three new problems by considering the question:"What if some or all of the given attributes are not true?" c. Thoroughly investigate one of the problems generated above.
3. (10 points)Find the maximum area of a trapezoid inscribed in a semicircle of radius 1.
Hint: Use the arithmetic mean-geometric mean inequality
a. Describe your solution.b. Discuss possible extensions.
Figure 3. Final assignment from a teachers' mathematics problem solving course modified for use with secondary students
Testing, unfortunately, often drives the mathematics curriculum. Most criterion referenced testing and most norm referenced testing is antithetical to problem solving. Such testing emphasizes answer getting. It leads to pressure to "cover" lots of material and teachers feel pressured to forego problem solving. They may know that problem solving is desirable and developing understanding and using appropriate technology are worthwhile, but ... there is not enough time for all of that and getting ready for the tests. However, teachers dedicated to problem solving have been able to incorporate problem solving into their mathematics curriculum without bringing down students' scores on standardized tests. Although test developers, such as the designers of the California Assessment Program, are beginning to consider alternative test questions, it will take time for these changes to occur. By committing ourselves to problem solving within our classrooms, we will further accentuate the need for changes in testing practices while providing our students with invaluable mathematics experiences.
Looking Ahead ...
We are struck by the seemingly contradictory facts that there is a vast literature on problem solving in mathematics and, yet, there is a multitude of questions to be studied, developed, and written about in order to make genuine problem solving activities an integral part of mathematics instruction.
Thus a teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. Polya (26, p. v.)
Further, although many may view this as primarily a curriculum question, and hence call for restructured textbooks and materials, it is the mathematics teacher who must create the context for problem solving to flourish and for students to become problem solvers. The first one in the classroom to become a problem solver must be the teacher.
James W. Wilson Maria L. Fernandez
Still Wondering About ...
The primary goal of most students in mathematics classes is to see an algorithm that will give them the answer quickly. Students and parents struggle with (and at times against) the idea that math class can and should involve exploration, conjecturing, and thinking. When students struggle with a problem, parents often accuse them of not paying attention in class; "surely the teacher showed you how to work the problem!" How can parents, students, colleagues, and the public become more informed regarding genuine problem solving? How can I as a mathematics teacher in the secondary school help students and their parents understand what real mathematics learning is all about?
Nelda Hadaway

math6

Problem Solving

An organization needs to define some standard of problem solving, so that leadership can effectively direct others in the research and resolution of issues.

In problem solving, there are four basic steps.

1. Define the problem

Diagnose the situation so that your focus is on the problem, not just its symptoms. Helpful techniques at this stage include using flowcharts to identify the expected steps of a process and cause-and-effect diagrams to define and analyze root causes.

The chart below identifies key steps for defining problems. These steps support the involvement of interested parties, the use of factual information, comparison of expectations to reality and a focus on root causes of a problem. What’s needed is to:

Review and document how processes currently work (who does what, with what information, using what tools, communicating with what organizations and individuals, in what time frame, using what format, etc).

Evaluate the possible impact of new tools and revised policies in the development of a model of “what should be.”

2. Generate alternative solutions

Postpone the selection of one solution until several alternatives have been proposed. Having a standard with which to compare the characteristics of the final solution is not the same as defining the desired result. A standard allows us to evaluate the different intended results offered by alternatives. When you try to build toward desired results, it’s very difficult to collect good information about the process.

Considering multiple alternatives can significantly enhance the value of your final solution. Once the team or individual has decided the “what should be” model, this target standard becomes the basis for developing a road map for investigating alternatives. Brainstorming and team problem-solving techniques are both useful tools in this stage of problem solving.

Many alternative solutions should be generated before evaluating any of them. A common mistake in problem solving is that alternatives are evaluated as they are proposed, so the first acceptable solution is chosen, even if it’s not the best fit. If we focus on trying to get the results we want, we miss the potential for learning something new that will allow for real improvement.

3. Evaluate and select an alternative

Skilled problem solvers use a series of considerations when selecting the best alternative. They consider the extent to which:

A particular alternative will solve the problem without causing other unanticipated problems.
All the individuals involved will accept the alternative.
Implementation of the alternative is likely.
The alternative fits within the organizational constraints.

4. Implement and follow up on the solution

Leaders may be called upon to order the solution to be implemented by others, “sell” the solution to others or facilitate the implementation by involving the efforts of others. The most effective approach, by far, has been to involve others in the implementation as a way of minimizing resistance to subsequent changes.

Feedback channels must be built into the implementation of the solution, to produce continuous monitoring and testing of actual events against expectations. Problem solving, and the techniques used to derive elucidation, can only be effective in an organization if the solution remains in place and is updated to respond to future changes.

Excerpted from G. Dennis Beecroft, Grace L. Duffy, and John W. Moran, The Executive Guide to Improvement and Change, ASQ Quality Press, 2003, pages 17-19.

math6

SEVEN STEPS TO PROBLEM SOLVING


"The message from the moon... is that no problem need be considered insolvable."
- Norman Cousins


There are seven main steps to follow when trying to solve a problem. These steps are as follows:
1. Define and Identify the Problem 2. Analyze the Problem 3. Identifying Possible Solutions 4. Selecting the Best Solutions 5. Evaluating Solutions 6. Develop an Action Plan 7. Implement the Solution
TQS Problem Solving Steps Problem Solving Techniques Frequently Asked Questions

1. Define and Identify the Problem
This first step is critical. It is essential for each group member to clearly understand the problem so that all energy will be focused in the same direction. A good way to define the problem is to write down a concise statement which summarizes the problem, and then write down where you want to be after the problem has been resolved. The objective is to get as much information about the problem as possible. It may be helpful to divide the symptoms of the problem into hard and soft data. Hard Data Includes: Facts, statistics, goals, time factors, history
Soft Data Includes: Feelings, opinions, human factors, attitudes, frustrations, personality conflicts, behaviors, hearsay, intuition
These steps may not always be pleasant, but after "venting" group participants may feel that the air has finally cleared and members can be more rational and cooperative.
Sometimes information needs to be gathered via various devices to define the problem. These devices may include: Interviews, statistics, questionnaires, technical experiments, check sheets, brainstorming and focus groups.
Develop a Problem Statement
It is essential to develop an objective statement which clearly describes the current condition your group wishes to change. Make sure the problem is limited in scope so that it is small enough to realistically tackle and solve. Writing the statement will ensure that everyone can understand exactly what the problem is. It is important to avoid including any "implied cause" or "implied solution" in the problem statement. Remember, a problem well stated is a problem half solved. State the Goal
Once the problem is defined, it is relatively easy to decide what the goal will be. Stating the goal provides a focus and direction for the group. A measurable goal will allow the tracking of progress as the problem is solved. Considerations
When defining the problem, ask the following:
Is the problem stated objectively using only the facts?
Is the scope of the problem limited enough for the group to handle?
Will all who read it understand the same meaning of the problem?
Does the statement include "implied causes" or "implied solutions?"
Has the "desired state" been described in measurable terms?
Do you have a target date identified?

2. Analyze the Problem

In this stage of problem solving, questions should be asked and information gathered and sifted. Do not make the mistake of assuming you know what is causing the problem without an effort to fully investigate the problem you have defined. Try to view the problem from a variety of viewpoints, not just how it affects you. Think about how the issue affects others. It is essential to spend some time researching the problem. Go to the library or develop a survey to gather the necessary information.
Questions to Ask When Analyzing the Problem:

What is the history of the problem? How long has it existed?
How serious is the problem?
What are the causes of the problem?
What are the effects of the problem?
What are the symptoms of the problem?
What methods does the group already have for dealing with the problem?
What are the limitations of those methods?
How much freedom does the group have in gathering information and attempting to solve the problem?
What obstacles keep the group from achieving the goal?
Can the problem be divided into sub problems for definition and analysis?
3. IDENTIFYING POSSIBLE SOLUTIONS Idea Generation Techniques


"The best way to have a good idea is to have a lot of ideas."
- Griff Niblack

Identifying possible solutions to the problem is sometimes referred to as finding "Optional Solutions" because the goal is to complete a list of all conceivable alternatives to the problem. Using a variety of creative techniques, group participants create an extensive list of possible solutions. Asking each group member for input ensures that all viewpoints will be considered. When the group agrees that every course of action on the list will be considered, they will feel some direct ownership in the decision making process. This may help put the group in the mood of generating consensus later in the decision making process.
You may already be familiar with some of these topics, but take the time to look through them anyway. The information you will find is valuable to your group's success.
Techniques Used in Solving Problems
These idea generation techniques are broken down into easy-to-follow steps that will help keep your group organized and on the topic at hand. We are basically giving you step-by-step instructions on how to accomplish each technique with ease and success.
Brainstorming Brainstorming is a problem solving approach designed to help a group generate several creative solutions to a problem. It was first developed by Alex Osborn, an advertising executive who felt the need for a problem solving technique that, instead of evaluating and criticizing ideas, would focus on developing imaginative and innovative solutions.
Steps
A group's members are presented with a problem and all its details.
Members are encouraged to come up with as many solutions as possible, putting aside all personal judgments and evaluations. "Piggy-backing" off another person's idea is useful.
All ideas are recorded so the whole group can see them.
Ideas are evaluated at another session. Characteristics
Procedure designed to release a group's creativity in order to generate multiple imaginative solutions to a problem.
Separates the idea-creation from the idea-evaluation process by not allowing any criticism to take place while the group is generating ideas.
May be more productive for each member to brainstorm quietly and then share ideas with the group (brainwriting).
Electronic brainstorming puts each member at a computer terminal and their ideas are projected to a screen so no one knows from whom an idea came.
Used by businesses and government to improve the quality of decision making.
Brainstorming for Teachers Brainstorming - Effectiveness in a Product Design Firm Buzz Groups
Steps
The facilitator presents a target question to the group.
If the group is large, divide into smaller groups (approx. six people).
Each group is given a copy of the target questions on an index card and a recorder/spokesperson is selected by seating. The individual then writes all ideas on index cards.
The group spends a few minutes thinking of and evaluating ideas.
The group reports its list to the entire assembly. Characteristics
A large group is subdivided into smaller groups which discuss an assigned target question, then report their questions back to the main group.
Encourages participation and involvement that is not feasible in large groups.
Technique can be used to identify problems or issues, generate questions to study, compile a list of ideas or solutions, or stimulate personal involvement.
Used by churches, schools, and company department heads to foster involvement in a large group assembly.
Suggestions for Team Buzz Sessions
Nominal Groups
Steps
The problem, situation, or question is stated clearly and concisely.
The coordinator asks participants to generate a list of the features or characteristics of the problem or question.
The coordinator gives the group five to fifteen minutes to work silently.
Each suggestion is recorded on a chart visible to all members.
Members clarify the items, but do not yet evaluate them.
Each person chooses his or her top ranked items.
The group engages in full discussion about the top rated items.
A decision is reached. Characteristics
Capitalizes on the finding that people working individually while in the presence of others sometimes generate more ideas than while interacting as a group.
Can enable members to reach a decision on a controversial issue without leaving a residue of bitterness from a win-lose conflict.
Members work individually in each other's presence by writing their ideas. They record these ideas on a chart, discuss them as a group, and finally evaluate them by a ranking procedure until members reach a decision.
Stifles effect of dominating members of the group.
Tendency for lazy members to let others carry the ball is minimized.
Adds structure to the brainstorming process. Delphi Methods
Steps
A Delphi Panel is selected by the facilitator.
The problem or issue is stated concisely in writing and sent to each of the Delphi panel for individual work.
The facilitator compiles another document that details all the individual positions taken by the panel and distributes a copy to each member
This procedure, with a facilitator compiling the individual comments into a single document and distributing it to the group, continues until a consensus is reached. Characteristics
Not a group decision technique.
Involves presenting a problem or an issue to the appropriate individuals, asking them to list their solutions, compiling a master list, circulating this master list to all participants, and asks them to comment in writing on each item on the list. The list with comments is then circulate to the participants. The procedure is continued until a decision is reached.
Good for when time and distance constraints make it difficult for group members to meet. Fantasy Chaining
Steps
Whenever the group is not talking about the here-and-now of the problem, it is engaged in fantasy.
Fantasy chaining is a group story-telling method wherein everyone in the group adds something to the topic at hand, which may not necessarily be the primary focus. Characteristics
Manifest theme is what the fantasy chain is about at the surface level.
Latent theme is the underlying theme (what the group members are really thinking about).
Helps the group define itself by creating symbols that are meaningful and that help determine its values.
Enables a group to discuss indirectly matters that might be too painful or difficult to bring out into the open.
Helps a group deal with emotionally "heavy" information.
Effective way in which groups create their shared images of the world, each other, and what they are about as a group.
A group's identity converges through these shared fantasies. Focus Groups
Steps
Instructor introduces a topic that is to be discussed by the group in any way they choose. Characteristics
Encourages unstructured thoughts about a given topic.
Often used to analyze people's interests and values.
Universities, large corporations, and political candidates use focus groups to understand how others perceive their strengths and weaknesses.
Focus Groups - Making it Work for You Behind a One-Way Mirror Metaphorical Thinking
A metaphor is a thinking technique connecting two different universes of meaning. The key to metaphorical thinking is similarity.
Excessive logical thinking can stifle the creative process, so use metaphors as a way of thinking differently about something. Make and look at metaphors in your thinking, and be aware of the metaphors you use. Metaphors are wonderful, so long as we remember that they don't constitute a means of proof. As by definition, a metaphor must break down at some point.
Steps
State the objectives of thinking in metaphors: to see comparisons between two ideas, and to gain new insights from comparisons
Brainstorm possible metaphors for some aspects of the problem
"Piggyback" on metaphors; build on them
Choose the best metaphors to carry further
Examine all imaginable areas of comparison in the metaphor
Ask questions the metaphor might answer
Look for insights into causes, effects, and solutions for your problem Considerations
When identifying possible solutions, it is essential to remember to:
Hold back from evaluating proposed solutions.
Make a point of "thinking outside" of your own experience and expertise.
Involve everyone in the process.
Go for quantity - at least 20 or so possible solutions before narrowing the list to between four and six of the best suggestions.

4. SELECTING THE BEST SOLUTIONS


Six Steps to Decision Making

"Decision making can be seen as bringing one into an ambivalent relationship
with both power and responsibility."
- Michael Lerner
Decision Making Techniques Teamwork - Decision Making Eight Steps to Decision Making
Define and Identify The Problem
The starting point in any conscious attempt at rational decision making must be the recognition that a problem exists. While effective meetings are essential to getting work done, most meetings leave us still looking for a decision. A good group meeting should bring the group members together. It should facilitate decision making, assist others in taking responsibility, and contribute to building team effort within the group.
The group begins with defining the problem. The group members only discuss the definition of terms and how the problem relates to other issues. Identifying the problem is very crucial. It is important to not define a problem so broadly that it generates never-ending questions.
Be Alert
Being alert at all times can help you work towards finding a solution in a logical manner. It is essential to be sensitive to morally charged situations. Everyone in the group must be alert and ready to make concise decisions if a problem were to arise. Group members should work through a series of steps designed to force them to reflect on certain aspects of a problem in a rational manner. Being alert is simply stating the obvious and immediate. For example, an auto accident, burnt out motor, and an overdrawn account.
Gather Information and Do Not Jump to Conclusions
This is the essence of the decision making process. It is very important to spend time on this step before suggesting solutions. It is said that successful groups do not jump to the solution stage quickly. They spend ample amount of time gathering information and analyzing the problem. The main purpose of this step is to gather as much information on a topic as possible. The group needs to think about their audience. They need to think about who will be reading it and when. While accuracy is important, there can be a trade-off between gathering information and letting morally significant options and information disappear.
State The Case Briefly
In this step, the group needs to come up with relevant facts and circumstances. They need to gather this information within the decision time available. It does not have to be a lot of information, but all of the important information needs to be stated in brief context.
What Decisions Have to Be Made?
Life is full of choices and decisions. Even deciding not to decide is, of course, a decision. This stage is very crucial to overcoming a problem, and, of course, making a group decision. The members of the group need to put thoughts and ideas into play in order to make good decisions. The group needs to brainstorm and gather lots of options to come to one final decision.
By Whom?
Remember that there may be more than one decision maker. Their interactions can be very important and influential in a group decision.
Specify Feasible Alternatives
In this step the group needs to begin defining the problem. The group members need to define the problem and come up with other ideas so they are not limited to just one final decision. If their first alternative does not work out, they can make another decision and choose other alternatives. The following are some steps to follow while finding feasible alternatives:
Define the problem. Try to phrase it as a question.
Discuss group goals and options.
Identify all options available at that time.
Determine the importance of the problem.
Make clear all meeting times and places.
State Live Options at Each Stage
In this step you should be sure to ask many questions. Each decision maker needs to take into account good or bad consequences. Here, you should ask what the likely consequences are of various decisions.
Identify Morally Significant Factors in Each Alternative
In this step you need to use your ethical resources to determine what the decision will be. The following are the most significant factors you should use as a guideline when determining your decision.
Principles
These are principals that are widely accepted throughout a group or organization. Respect Autonomy
Members of the group need to ask themselves questions such as, Would I be exploiting others? Have promises been made? Don't Harm
After making the decision, think to yourself whether or not your decision will hurt anyone. Be Fair
Be fair and work willingly with the members of your group.
Use Good Context
A decision must be made, but which solution should your group choose? In identifying the best solution or solutions to the problem, the group should consider from among the four to six suggestions that were decided upon from the ideas which they had gathered. The different factors, or criteria, that people use to make their decisions are often unclear or never voiced. This can lead to misunderstandings and misinterpretations of other people's motives. Discussion of Possible Resolutions
Gathering Ideas and Information on the Actual Problem
Determine who exactly is your audience. This allows you to specify a solution that best addresses a specific audience.
Research and establish the history of the problem to be solved, as well as what caused the problem to occur. This allows for accuracy within your solution.
Discuss how the problem to be solved relates to other issues. However, be careful not to bring forth any other problems while solving the initial problem.
Analyze and examine the facts and all of the gathered information. This allows the group to challenge facts and assumptions, making sure they can withstand any type of scrutiny or disagreement.
Make sure that you have gathered enough information on the problem. Establish Some Type of Decision Criteria
The discussion of the group should focus on what makes an acceptable decision.
Examine what an ideal decision consists of and what should be included and excluded out of that decision.
Discuss what a reasonable or fairly good solution would be. This becomes important when the ideal solution can not be reached.
Decide what standards the group should utilize to judge a decision.
Consider what is valid and feasible about the decision made. Discuss Possible Solutions
Address such questions as:
Have all solutions been accounted for?
What, if any, is the evidence to support each of the chosen decisions?
Did the group use brainstorming techniques to produce ideas?
Selecting or Determining the Best Solution
At this stage in the process the group is working towards an agreement on the final solution. This is done by testing all previously made solutions using the decision making criteria set forth by the group. The group goal in this step is to make sure they have found the solution that will best solve the problem and address any other issues that may have been a consequence of that problem.
During this phase, the group should eliminate any solution that does not meet the requirements and focus on those that could ultimately be utilized. The group should be concerned with whether or not the solution chosen solves the problem or just minimizes it.
Is the solution workable in relation to the problem?
Are there any limits that the solution presents?
When looking at the advantages and disadvantages, which are there more of?
Does the chosen idea live up to the standards of the decision criteria?
Are the facts and information gathered consistent with the proposed solution? Implementing the Solution
In this phase the group should be focusing on two main goals.

The best way to make the solution apply and function when applied to the problem.
What resources are needed for the solution to work? For this to work, complete dedication on the part of all group members is needed. Everyone in the group has to be willing to work with one another while offering their unique skills and talents. Group members also have to be willing to take full responsibility for the solution they choose.
Accountability within the group plays a very important part in the decision implementation process. While in this phase, the group should ask such questions as (Schein, 1969):
What do we have to do to accomplish our proposed course of action?
Who will be responsible for implementing the proposed plan?
When can our group reasonably expect results?
What unplanned events or accidents are likely to jeopardize our actions?
What people should we consult who can help us with our proposal?
What people should we consult who could threaten our proposal? Once this is done, the group can implement the solution.
During this final step, it is important to have the support of the entire group. Keep in mind that in case this primary solution does not work as planned, the group will have to look for alternative solutions. Tracking the effect of the solution in the long run also serves to be a helpful future model and determines what is and what is not needed in a solution.


5. Evaluating Solutions
There are several ways to evaluate the chosen solutions, and writing them all down will help the group to choose the best solution to the problem.

Making a T-Chart to Weigh the Pros and Cons of Each Idea
It is often helpful to make a T-chart and ask members of the group to name the pros and cons of each solution. This method will visually illustrate the strengths and weaknesses of each solution.
Develop and Assign Weights to Criteria
The key to avoiding possible deadlock in the decision making process is to put all criteria people are thinking about on the table. This way, all group members are clear as to what criteria others are using.
Prioritize the Criteria
The next step is for the group to agree on how important these criteria are in relation to each other. For example, is cost the most important criterion, or low resistance by others, etc. The criteria should then be rated in terms of importance. Assign a number to each criterion so that all criteria together total 100. Rate Proposed Solutions Using Criteria
Using the four to six possible solutions, score (on a scale of 1 to 10) each solution against each criteria. Repeat this for each criterion. Multiply this score to the weighting, then add the weighted scores for each solution. This exercise will help you to compare alternatives objectively. Considerations
What are the advantages of each solution?
Are there any disadvantages to the solution?
Do disadvantages outweigh advantages?
What are the long and short-term effects of this solution if adopted?
Would the solution really solve the problem?
Does the solution conform to the criteria formulated by the group?
Should the group modify the criteria?


6. Develop an Action Plan
An action plan is a chart that lists the tasks that need to be done and identifies who will be responsible for each, when and what action is necessary, where to start, and how.
Divide the Solution Into Sequential Tasks
Looking at your solution as one task may seem too great an undertaking. It is much more productive to divide it into sequential tasks which act as measurable steps toward the solution. When dividing the solution into tasks, be sure to include a timeline, what is to be done, and who will do it.
Develop Contingency Plans
The best laid plans of mice and men... Even the best of plans get stalled, sidetracked, or must be changed midstream because of something unforeseen. Most times these circumstances cannot be prevented, but you can and should prepare for potential kinks by having a contingency plan. Having such a plan will keep the momentum going instead of having to stop and figure out what to do when an unplanned event occurs. Action Planning Question Checklist
The following checklist will be helpful to ensure that all bases are covered in your plan of action:
What is the overall objective and ideal situation?
What is needed in order to get there from here?
What actions need to be done?
Who will be responsible for each action?
How long will each step take and when should it be done?
What is the best sequence of actions?
How can we be sure that earlier steps will be done in time for later steps which depend on them?
What training is required to ensure that each person knows how to execute each step in the plan?
What standards do we want to set?
What resources are needed and how will we get them?
How will we measure results?
How will we follow up each step and who will do it?
What checkpoints and milestones should be established?
What are the make/break vital steps and how can we ensure they succeed?
What could go wrong and how will we get around it?
Who will this plan affect and how will it affect them?
How can the plan be adjusted without jeopardizing its results to ensure the best response and impact?
How will we communicate the plan to ensure support?
What responses to change and other human factors are anticipated and how will they be overcome? Considerations
Have you considered what resources will be needed?
Have you developed contingency plans for the most critical action steps?
Are the necessary people aware of the contingency plan?


7. Implement the Solution
Sometimes the groups who choose the solution are not the ones who will implement it. If this is the case, members who select the solution should clearly explain why they selected it to the ones who will implement it. Showing that the problem solving process was an organized and orderly process will convince others that the solution is valid.
Monitoring
A designated member of the group should monitor whether or not specific tasks are being performed or short-term targets are being achieved as planned. This monitoring should take place regularly until all tasks are completed. Some suggested monitoring techniques are:
Tickler file
Compliance reports
Group meetings
Individual meetings
Customer/user interviews
Surveys and written questionnaires
Quality control spot checks
Audit
Walk through or role play
Trend graph
Checkpoints on action plan
Personal inspection of all work
Budget controls
Grapevine Implement Contingency Plan if Necessary
As conditions change during monitoring and evaluation of the Action Plan, it may become necessary to implement the contingency plans to continue moving toward the goal.
Try making a three-column chart for your contingency plan listing:
What could go wrong?
How can you prevent this from happening?
How will we fix it if it does happen?
Evaluate Results
This step may involve repeating the initial seven-step problem solving process to address additional problems as needed. Make certain that the goal has been reached and that a plan is in place to ensure that the problem will not recur. Group Problem Solving
Ask the following questions and score each answer on a scale of 1 (no participation at all) to 5 (participated very well) to ascertain how well your group solved the problem:
How well did the group assess the problem or decision?
How well did the group identify its goal?
How well did the group identify the positive consequences of the solutions under consideration?
How well did the group identify the negative consequences of the solutions under consideration?
Did the group draw reasonable conclusions from available information?
Considerations
What role will others play in evaluating progress during implementation?
How will you know if the implementation is on track?
How will you know when to implement the contingency plan?
Who makes the decision?
Has the goal been reached?
Are plans in place to ensure the problem does not recur?
A Recipe for Problem Solving
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