Problem solving is a key component of DISCOVER, forming a basis for both the Assessment and the Curriculum Models. But the DISCOVER approach to problem solving is different. It is built upon the premise that considering a problem’s structure and method (how it is solved) is as important as actually finding the solution. Each is briefly summarized below:
Problem Structure—The way a problem is formed and presented is critical to its effect on learning. An effective use of problem solving must include problems that vary widely. DISCOVER rates Problem Structure on a scale that ranges from what we call a “Type I” problem, up to a “Type V” problem. A Type I problem is highly structured and closed whereas a Type V problem is completely open-ended and complex. All conceivable problems fall somewhere in the continuum between the two extremes. For sake of illustration, we break the continuum down into Problem Type I, Type II, Type III, Type IV, and Type V.
Problem Method—The way a problem is solved will depend on its form and presentation, as discussed above. A Type I problem can be solved in only one way and the solver simply must know the right method, to arrive at the right solution. A Type V problem, on the other hand, might have an infinite number of ways to arrive at a solution and the solver must determine which method(s) might be better than others and if any given method is even appropriate…with the possibility that there are no appropriate methods because there is no definitive solution. Consideration of Problem Method is an important part of developing critical thinking skills.
Problem Solution—The Type I problem has a single right answer (ex. 1+1=2). A pure Type V problem is so abstract that it might have an infinite number of possible solutions or it might not have a solution at all. Arriving at a “right” answer for a Type V problem is usually highly subjective, depending on what method was chosen and on how the initial problem was interpreted in the first place (ex. “Does God exist?”).
Figure 1 below illustrates what we call the “DISCOVER Problem Continuum” (Maker and Schiever, 1991), a tool used in assessment and curriculum. The five Problem Types are displayed, along with how much information is known—how much structure is provided—for both the problem presenter and the problem solver in each Problem Type. A detailed analysis follows below. The DISCOVER Problem Continuum actually is an expanded version of a model developed by Getzels and Csikszentmihalyi (1976,1967). Maker and colleagues added Problem Types III and IV, to provide a more fluid transition between the Types, based on data observed in their research (C. June Maker, 1993; 1978; 1981; Whitmore & Maker, 1985). The DISCOVER Assessment and all DISCOVER Curriculum instruments use all of the Problem Types.
Figure 1: DISCOVER Problem Continuum
Problem
Type Examples
Type 1: The problem is simple, closed, and known to both the presenter and the solver. The method, also, is known to the presenter and to the solver. The solution is known to the presenter but must be derived by the solver. (Example: “Solve 3 + 4.” The problem is clearly defined. The method is addition, and the solver need only determine that “7” is the correct answer.)
Type 2: The problem is simple, closed, and is known both to the presenter and the solver. The presenter knows the method and solution, but the solver must derive both. (Example: “If a cookie jar contains ten cookies and you eat two of them, now how many cookies are in the jar?”—The problem is clearly defined. The solver must determine that the method is subtraction and that the correct answer is 8).
Type 3: The problem is known to both the presenter and the solver, but is more open and complex. There can be a range of correct methods and solutions, all of which are known to the presenter but must be derived by the solver. (Example: “Use these numbers to write as many correct equations as possible [3,5,2]”—When used with young children, there are only four answers that fall within a reasonable scope of experience [3+2=5, 2+3=5, 5-3=2, 5-2=3], qualifying this problem as Type 3. The solver, in determining the methods, must see the possibility of using different operations and number orders to arrive at the maximum quantity of answers.)
Type 4: The problem is known to the presenter and to the solver, but the methods and solutions are unknown to both. (Example: “You need to get to the other side of a large stream. What do you think might be the best ways to do so?”—Several categories of methods and solutions might be given: through the air [fly, jump, swing across on a vine]; floating [in a boat, on a log]; walk across [on a bridge, downed tree, or long board], or any number of creative and perhaps unexpected answers.) The Type 4 problem has a clear goal but not necessarily a right answer, requiring the solver to gather as much information as possible and analyze the potential methods and solutions.
Type 5: The problem(s), method(s), and solution(s)—all of which are completely open-ended and complex—are ill defined, for both the presenter and the solver. (Example: “What are the most important problems facing humanity and what should be done about them?”—The first task is to start with the problem, itself, and determine if there really is a problem and if so, what is it (or what are they). After isolation of a specific problem, methods and solutions can be evaluated. The process may be open to interpretation from start to finish and the final results will vary widely depending on the perspective, preconceptions, analysis and intent of both the presenter and the solver.
DISCOVER researchers are convinced that problem solving should play a key role in education, at all levels, precisely because it is so important in everyday life. Each of us may make hundreds of decisions every day, and the vast majority of these decisions fall in the range between a Type 3 and a Type 5, with most being closer to a Type 5. This fact is surprising to many people, until they think about it. Most decisions we make are not simple or closed because there are numerous possible variations or alternatives and there are myriad intended (or unintended) consequences to be considered. Getting dressed, for example, typically is a Type 4 problem; usually there is a clear need to do so, yet the method and solution will vary according to one’s state of mind, habit, the weather, the day’s planned events, and so forth. Getting dressed becomes a Type 5 problem on days when you are sick, or do not have specific plans, and you begin the day by asking, “Do I really need to get dressed or should I just stay in bed?”
DISCOVER was created because of a simple realization…that if real-life problems typically are in the Type 4 and Type 5 range, does it not make sense that we should test students’ ability to make these kinds of decisions? Likewise is it not imperative for educators to actually help students develop their problem solving skills, especially regarding complex, open-ended problems? Unfortunately much remains to be done in the standard classroom. Virtually all traditional testing methods, and most traditional lesson plans, continue to be designed around Problem Types 1 and 2 only—drill and practice, find-the-right-answer kinds of approaches. One individual observed that we spend twelve-plus years teaching our students how to solve Problem Types 1 and 2, and then expect them to function in the real world where these Problem Types are virtually non-existent.
Programs for gifted students usually are better, and are more likely to include all of the Problem Types. But even here a bizarre and little known situation occurs that can adversely affect a child’s development. The scenario is as follows: In most schools, superior performance on traditional, standardized tests scores is the predominant selection criterion used in programs for the gifted. Therefore because the questions on these tests contain only Problem Types 1 and 2, students who solve Problem Types 1 and 2 best are chosen. Then they are thrown into a track that emphasizes, instead, Problem Types 3, 4, and 5, causing many to feel lost and frustrated. At the same time, those students who did not do as well on the standardized tests, perhaps because they are more comfortable with, and better at, solving Problem Types 3, 4, and 5, are placed back in the standard curriculum that contains almost entirely Problem Types 1 and 2, causing them to feel frustrated and bored. DISCOVER researchers believe that the only way to remedy this mismatch is to determine which individuals can solve which Problem Types best (and in which areas), so that the curriculum is developmentally appropriate.
For those curious about how commonly used assessment methods and programs stack up, in terms of which Problem Types they include, see Figure 2 below.
Figure 2: Common Problem Types
We acknowledge that changing the overall situation is complicated and will take time. It will require a systematic refocusing of perspective, such that testing and teaching methods begin to emulate real-world problem solving. We realize, too, that inclusion of Type 4 and Type 5 problems in testing and teaching is viewed as problematic because of the extra time required and the difficulty of “grading” such products. But we as educators must begin to ask tough questions. Are we adequately supplying our students with the skills they need to become successful adults? Are we testing what needs to be tested?
Some studies suggest, no. Scholars who have completed critical reviews of research (Baird, 1985; Hoyt, 1966; Nelson, 1975; O’Leary, 1980; Reilly & Chao, 1982; Wallach, 1976), including meta-analysis (Samson, et. al., 1984; Cohen, 1984), have concluded that although achievement and abilities test scores can predict future academic success (e.g. grades and later achievement test scores), their relationship to adult success factors is minimal. O’Leary (1980), for example, reports that the average correlation between measures of academic success and indicators of adult career success is .17. In other words, we are training our students how to pass tests but not necessarily how to succeed in life. Furthermore, reviews of research on problem solving (Rogers, 1993) have revealed that very few studies have ever included Type IV or V problem situations. As a result, little has been learned about the abilities and skills actually needed in the structuring and solving of real-world problems. Is it any surprise that teachers often are not sure how to teach the skills?
But progress is being made, both within DISCOVER and without. Tests of creativity, for example do tend to include more emphasis on complex, open-ended problems, and research on the predictive validity of these instruments is more encouraging. Only a few studies are available; the most notable is Torrance’s twenty-two-year follow-up of 211 individuals whose mean age was 27.5 years. All had been assessed with the Torrance Tests of Creative Thinking (TTCT) as elementary school students. Using a creativity index made up of scores on the TTCT over a three year period, and a variety of indicators of adult creative achievements, he found a correlation of .58 (p<.001) (Torrance, 1981). In a well designed reanalysis of Torrance’s data, Yamada and Tam (1996) found the four predictors—the creativity index, childhood future career image, IQ, and the presence of a mentor—explained 49% of the variance in adult creative achievement. The standardized regression coefficients were .44, .24, .20, and .16 respectively. The results of Torrance’s research are more conclusive than others, but the trend can be seen in numerous studies (c.f. Howieson, 1982; Baird, 1985).
In general, reviewers and authors conclude that the more the predictor (e.g. test or behavior such as high school or college achievements) resembles the criterion to be predicted (e.g. another test, grades, behaviors such as adult achievement), the more successful the prediction. For example, measures of creative writing predict later publications of fiction better than measures of general ability; the advanced knowledge sections of the GRE predict productivity in physics, geology, engineering and chemistry better than general verbal and quantitative sections; and student achievements in high school (e.g. publications, winning science fair competitions) predict later accomplishments—better than grades, academic achievement, or ability tests.
If we are to realize the vision articulated in the new curriculum standards and prepare a generation capable of designing new approaches to new problems, we must assess and develop a wide range of problem solving abilities. Instead of measuring and developing “decontextualized” knowledge or divergent thinking, we must measure the production, flexible use, originality, and addition of richness and detail in the solving of complex, realistic problems in specific academic and intellectual domains. This is the intent and promise of the DISCOVER assessment. We also must develop those abilities in real-life problem solving situations. This is the intent and promise of the DISCOVER curriculum.
References
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