e. c u d o r t rep and Critical Thinking o n CHAPTER 13 Transfer, Problem Solving, o d only, 9-03 e s u l a -0 5.ncu.edu n 3 o 2 s r 0 e 2 P o36 @ 7 3 3 Transfer nino1 . m Types of Transfer Theories of Transfer Factors Affecting Transfer uce. d o r p e not r o d , y l on e s -03 cu.edu u 9 l 0 a n 3 o 2 s Cognitive FactorsP Affecting Solving 0 er Successful2Problem 5.n 6 3 o @ 7 Problem-Solving Strategies nino133 . m Problem Solving Theories of Problem Solving Meaningless versus Meaningful Problem Solving Facilitating Transfer and Problem Solving in Instructional Settings ce u d o r p ot re n o d , only 9-03 Fostering Critical Thinking in the Classroom e s u l a rson 023-0o365.ncu.edu e 2 P Summary 7@ 3 3 1 o n m.ni Critical Thinking Developmental, Individual, and Cultural Differences in Critical Thinking . LEARNING OUTCOMES 13.1. Describe various forms that transfer might take and the conditions in which productive transfer is most likely to occur. e. c u d o r t rep o n o d ly, of problem 13.2. Contrast behaviorist and cognitivist solving, and identify key cognitive factors onviews e 3 s 0 u 9 l a -0 5.ncu.edu n 3 o affecting problem-solving success. 2 s r 0 e 2 P o36 @ 7 3 3 13.3. Distinguish between.algorithms nino1 and heuristics in problem solving, and describe several general m heuristics that have broad applicability to different kinds of problems. 13.4. Identify effective strategies for facilitating transfer and problem solving in instructional settings or with contemporary digital technologies. uce. d o r p e not r o d , y l e on -09-03 s u l du a e . n 3 u o 2 c s r 0 n . e 2 P o365 @ 7 3 3 1 m.nino 13.5. Describe several forms that critical thinking might take in different content domains, and identify strategies for helping students critically evaluate various sources of information both in print and online. m In my many years of college teaching, the course I taught most frequently was an undergraduate educational psychology course for students preparing to become teachers—a course that included such topics as human memory, operant conditioning, child development, motivation, classroom management, and e. c u d o r t rep o n o d and exams. But I’ve always wondered whether the thingsn ly,students03learned in my class later influenced o that e s u l a use psychological how they taught their own students. Could they of learning, educhild development, -09concepts . n 3 u o 2 c s r 0 n . e 2 5 P 6 of educational psychology help o3theories and motivation to help them plan good lessons? Did principles and @ 7 3 3 1 them solve instructional problems? m.nino good assessment practices. Most students seemed to master the material pretty well: They made insightful contributions to class discussions, and they could apply what they’d learned as they completed assignments Another topic I addressed in my educational psychology course was the importance of psychological and educational research for informing instructional practice, and throughout the semester I regularly alerted students to research findings on various topics. But after my students finished their formal training and uce. d o r p e not r o d , y l onimportant9-or,0instead, eand 3 s u researchers’ conclusionsnwere valid completely l du bogus? 0 a e . 3 u o 2 c s r 0 n . e 2 P 365 o @ 7 3 The questions I’ve posed in the first paragraph are questions about transfer and problem solving; those in 13 o n i n . the second paragraph m are concerned with critical thinking. In this chapter we’ll look at all three processes became teachers, they needed to keep abreast of new research results. Could they distinguish between well-designed studies and those that were poorly conceived and executed? Could they determine whether and consider how we might promote them in classroom settings. ce. u d o r p ot re n o d , only 9-03 e s u u l a 2023-0o365.ncu.ed Person 7@ 3 3 1 o n m.ni Transfer uce. d o r p e ot r nPeople o d , y occurring. Transfer is a regular part ofo everyday life: continually encounter new situations and draw l n e 3 s 0 u 9 dutransfer is an essential component of al and e -0deal with . on previously acquired knowledge skills to them. In fact, n 3 u o 2 c s r 0 n . e 2 5 from scratch about how to behave in every new P o3to6start human functioning. Without it, people would have @ 7 3 3 o1 of their time in trial-and-error learning. .ninmuch circumstance and would mspend When something you learn in one situation affects how you learn or perform in another situation, transfer is If you think back to the discussions of classical and operant conditioning in Chapter 3 , you might realize that we’ve already talked a bit about transfer. In that chapter we considered the phenomenon of generalization: After an organism learns a response to one stimulus, it often makes the same response to similar stimuli—stimuli it hasn’t necessarily encountered before. uce. d o r p e not r o d , y l e on -of0history s -03is completely u 9 l a eduuseless; as one high school . household budget. Many people think that knowledge n 3 u o 2 c s r 0 n . e 2 5 P o36daily life. When do you have a chance to use student put it, “History is a subject that is not related to your @ 7 3 3 1 And some teachers reinforce inappropriate behaviors in .nino2011). the gone and dead knowledge?” m(Hsiao, Transfer should typically be a top priority in instructional settings: Students should be able to apply what they’ve learned to situations and problems in their out-of-school lives. Yet the things people learn in school don’t always transfer to new situations. Some people don’t use basic mathematics to plan a realistic their classrooms, ignoring the behaviorist principles they learned in their college psychology classes. Many learning theorists agree that, in general, much school learning seems to yield inert knowledge that students never use outside the classroom (e.g., M. Carr, 2010; Haskell, 2001; Levstik, 2011; Whitehead, 1929). Types of Transfer e. c u d o r t rep o n o d , declarative onlyother e piece of declarative knowledge is helping youlrecall several When your skill in s -03 ctidbits. u u 9 d 0 a e . n 3 u o 2 s 0 procedural n throwing a baseball helps you cast is assisting you in Pear fishing line, your2existing 65.knowledge 3 o @ 7 3 learning a new procedure. When your understanding nino1of3 the base-10 number system helps you as you . m “borrow” in subtraction problems, your declarative knowledge is guiding your execution of a procedure. Transfer can involve declarative knowledge, procedural knowledge, or an interplay between the two. For instance, when the word HOMES helps you remember the names of North America’s five Great Lakes, one Transfer can go in the reverse direction as well—from procedural to declarative knowledge—as I found out a few years ago when I tried to use my waterskiing skills. I had water-skied quite a bit in my teens and twenties and fancied myself to be a reasonably good skier. After a long hiatus, I tried skiing again when I was in my mid-fifties. The boat had a hard time pulling me up and had to go quite fast—too fast for my comfort zone—to keep me on the water’s surface. I had, alas, gained 30 pounds during that ski-free hiatus. The new declarative knowledge I acquired in the experience—more poundage requires more speed to stay e. c u d o r ep intake. afloat—spurred a renewed commitment to minimize my junk t rfood o n o d ly, oncan e 3 kinds s u 9-0different l In addition to acknowledging that transfer involve 0 a edofuknowledge, theorists have made . n 3 u o 2 c s r 0 n . e 2 5 P among types of transfer:@ o36versus negative, vertical versus lateral, near versus far, several distinctions positive 7 3 3 1 and specific versus general. m.nino Positive versus Negative Transfer When learning in one situation facilitates learning or performance in another situation, positive transfer is at work. For example, practice in reading helps with spelling, and vice versa (N. J. Conrad, 2008). Two longterm memory storage processes we’ve come to know well—meaningful learning and elaboration—are also instances of positive transfer, because they involve using previously acquired information to understand and uce. d o r p e ot r n o d , y l material is attached, helping a learner fill in holes ideas are ambiguous or incomplete, or onwhen new e 3 s 0 u 9 l du 0 a e . n 3 providing a concrete analogy that makes abstract ideas easier to.understand. u o 2 c s r 0 n e 2 P 365 o @ 7 3 3 hinders a person’s ability to learn or perform in a o1situation In contrast, when something learned m.nininone remember new material (Brooks & Dansereau, 1987). “Old” information can help meaningful and elaborative learning in a variety of ways—perhaps by serving as a conceptual framework to which new second situation, we have a case of negative transfer. For example, when driving a car with an automatic transmission, people accustomed to driving a standard transmission may step on a clutch that isn’t there. People who learn a second language typically apply patterns of speech production characteristic of their native tongue, giving them a foreign accent; they may also mistakenly apply spelling patterns from their native language (Fashola, Drum, Mayer, & Kang, 1996; R. A. Schmidt & Young, 1987; Sun-Alperin & Wang, 2008). Students accustomed to memorizing facts in other college courses often don’t perform well on my own application-oriented exams. uce. d o r p e r not may o d , y l Negative transfer often rears its ugly head in work with o decimals: Students erroneously apply n e 3 s 0 u 9 l du 2005). For 0 2010; a numbers2(Karl e . n 3 mathematical rules they’ve learned for whole &-Varma, Ni & Zhou, u o 2 c s r 0 n . e P o365 @ example, when asked to compare decimals such as these: 7 3 3 1 m.nino 2.34 versus 2.8 students may apply the rule “More digits mean a larger number” and conclude that 2.34 is the larger one (Behr & Harel, 1988). Another rule inappropriately transferred to decimals is this whole-number rule: “When a number is divided, the result is a smaller number.” Even college students show negative transfer of this rule; for instance, many assert that the answer to this problem: e. c u d o r t rep o n o d nly, 1990). e& oGraeber, s -03The canswer u is a number smaller thann5a(Tirosh 9 l 0 edisuactually about 7.69, a larger number. . 3 u o 2 s r 0 n . e 2 5 P o36thing and sometimes not. @ 7 3 As you can see, then, transfer iso sometimes a good 3 1 m.nin 5 ÷ 0.65 Vertical versus Lateral Transfer In some subject areas, topics build on one another in a hierarchical fashion, so that a learner must master one topic before moving to the next. For example, an elementary school student should probably master principles of addition before moving to multiplication, because multiplication is an extension of addition. Similarly, a medical student must have expertise in human anatomy before studying surgical techniques: It’s . duce o r p e r not o d , y l e on s u l a n so 9 03 d hard to perform an appendectomy if you can’t find an appendix. Vertical transfer refers to such situations: A learner acquires new knowledge or skills by building on more basic information and procedures. l us -03 cu.edu 9 0 3 2 20 isn’toessential 65.nfor learning English, yet knowing Spanish prerequisite to the second. Knowledge of Spanish 3 @ 7 13are3similar in the two languages (e.g., Proctor, August, Carlo, o can help with English because many words n i n . m & Snow, 2006). When knowledge of the first topic is helpful but not essential to learning the second one, a Person In other cases, knowledge of one topic may affect learning a second topic even though the first isn’t a lateral transfer is occurring. Near versus Far Transfer Near transfer involves situations or problems that are similar in both superficial characteristics and ce. u d o r p re per hour within 5 seconds. otmiles An automotive engineer has designed a car that can reach a speed ofn50 o d , y onl 9-03 e s What is the car’s rate of acceleration? u l a -0 5.ncu.edu n 3 o 2 s r 0 e 2 P 36formula v = a × t (velocity = o @ 7 Let’s assume that you’ve learned how to solve this problem using the 3 13 m.ninothis problem: acceleration × time elapsed). You then encounter underlying relationships. For example, consider the following problem: A car salesperson tells a customer that a particular model of car can reach a speed of 40 miles per hour within 8 seconds. What is the car’s acceleration rate? The two problems have similar surface characteristics (both involve cars) and similar underlying structures (both involve the relationship among velocity, acceleration, and time). But now imagine that after solving the uce. d o r p e not rcan increase its speed by 6 kilometers-per-hour o A zoologist reports that a cheetah she hasly been observing d , n ocheetah e s 03a running u l each second. How long will it take the to9reach speed du of 60 kilometers-per-hour? 0 a e . n 3 u o 2 c s r 0 n . e 2 P 365 o @ 7 3 Although the general structurenisothe i 13same as before (once again, the v = a × t formula applies), we’ve now n . m switched topics (from cars to cheetahs), units of measurement (from miles-per-hour to kilometers-perfirst problem, you instead encounter this problem: second), and unknowns (from acceleration rate to time). Far transfer involves two situations that are similar in one or more underlying relationships but different in their surface features. An even “farther” instance of transfer might involve taking the v = a × t formula outside the classroom altogether—say, to real-world problems on a boat or at a race track. Specific versus General Transfer uce. d o r p e not r o d , y l transfer task overlap in some way. For example, about3human anatomy should help a veterinary e onknowing s -0 cu.anatomical u 9 l du features. A student who 0 a e n 3 o student learn dog anatomy because the two 2 species have many .parallel 2 s r 0 n e P 65 or Chinese because English and German o3Navajo @ knows English should more easily learn German than, say, 7 3 3 1 inosyntactical m.nmany have many similar words and share features. Both near and far transfer are instances of specific transfer, in which the original learning task and the In general transfer, the original task and the transfer task are different in both content and structure. For example, if knowledge of Latin were to help a student learn physics, or if the study habits acquired in a physics course facilitated the learning of sociology, then general transfer would be occurring. e ce. u d o r p t rD.e Gray & Orasanu, 1987; oW. (S. M. Barnett & Ceci, 2002; Bassok, 1997; Di Vesta & Peverly,o1984; n d , onlygeneral 3 occurs Perkins & Salomon, 1989). In fact, the question ofewhether at s u uall has been the 9-0transfer l d 0 a e . n 3 u o 2 c s 0 r n . subject of much debate over the early3 and theories of transfer, Peyears. We turn now2to both@ 65contemporary o 7 3 which vary considerably in their views about what things 13 transfer and when. m.nino Near transfer is more common than far transfer, and specific transfer is more common than general transfer Theories of Transfer How does transfer occur? We’ll look first at an early view of transfer—one that predates twentieth- and twenty-first-century learning theories—and then see what both behaviorists and cognitivists have had to say about how and when transfer takes place. ce. u d o r p t re odifficult n o d In days of yore, serious scholars often studied rigorous, topics—for instance, Latin, classical Greek, , y nl o e 3 s 0 u u subject areas may have had no 9today. n las frequently d 0 and formal logic—thatoaren’t studied Although these a e . n 3 u 2 c s r 0 . 2 Pe 365 that mastering them would improve performance in o specific applicability to everyday tasks, scholars believed @ 7 3 no13as the middle of the twentieth century, students had regular practice in nirecently many aspects of daily life. m.As A Historical Perspective: Formal Discipline memorizing things—poems, for example—apparently as a presumed means of improving their general learning capabilities. Such practices reflect the notion of formal discipline: Just as you exercise muscles to develop strength, you exercise your mind to learn more quickly and deal with new situations more effectively. The formal discipline perspective emphasizes the importance and likelihood of general transfer, the idea being that learning in one situation improves learning and performance in another situation regardless of how different the two situations might be. But as researchers began to study human learning systematically, uce. d o r p e they soon discarded this mind-as-muscle idea. For example, n ino research t r he described in 1890, psychologist o d , y 1 nlover the-course William James memorized a new poem s each of several weeks, predicting that the oday e 3 0 u 9 l duimprove; if anything, he 0 a e . n 3 u o 2 c s r 0 n practice would enhance his poem-learning ability. Yet his poem learning didn’t . e 2 5 P 36more o @ 7 3 learned his later poems more slowly thanothe early ones. And recently, researchers have found that 13 n i n . m skill requiring precise, detailed thinking about a logical sequence of learning computer programming—a events—has no impact on logical thinking in areas unrelated to computer use (Mayer & Wittrock, 1996; Perkins & Salomon, 1989). 1 You previously encountered William James in Chapter 7 . More than 100 years ago, James (1890) suggested that human memory has three components very similar to those in the now-popular dual-store model of memory. uce. d o r p e not r o d , y l n ofindings e 3 s -0emerged u that I said most contemporary theorists. Some intriguing have lately 9 l duto suggest that 0 a e . n 3 u o 2 c s r 0 n . e 2 general mental exercise might P indeed have long-ranging transferoeffects. 365 For example, in research that was @ 7 3 13 Snowdon (2001) studied a convent of elderly nuns widely publicized in the popular media at the inoDavid .ntime, m who kept mentally active with seminars, debates, puzzles, journal writing, and similar activities well into their The consensus of most contemporary theorists is that general transfer, in the extreme sense portrayed by the formal discipline perspective, probably doesn’t occur (e.g., Haskell, 2001; Salthouse, 2006). But notice nineties. Many of the nuns generously donated their brains to science, and postmortem studies found many more axons and dendrites than are typically found in 90+-year-olds. Snowdon’s research didn’t conclusively demonstrate causation—perhaps the convent attracted especially bright nuns to begin with—but other recent studies have shown that certain kinds of mental exercise do seem to have broad benefits. For example, when children have practice using a computer joystick to rapidly move a cartoon cat to various ce. u d o r p ot re n o d , y onlmemory e 03 seems s u daily practice with simple computer-based u memory in other, dissimilar 9-tasks l 0 a .etodenhance n 3 u o 2 c s 0 r n 2 . e 5 P situations, at least in the short run (Jaeggi, Buschkuehl, o36 Jonides, & Perrig, 2008). Personally, I’m taking a @ 7 3 3 1 wait-and-see attitude about .nallinofothis: Such findings need to be replicated before we jump to definitive m conclusions about the value of general mental exercise. locations on a computer screen, their enhanced attention skills transfer to very different situations, presumably reflecting enhanced central-executive skills (M. I. Posner & Rothbart, 2007). And for adults, An Early Behaviorist Theory: Thorndike’s Identical Elements Let’s consider the work of another early psychologist: Edward Thorndike, whose cat-in-a- puzzle-box observations laid a foundation for behaviorist ideas regarding reinforcement and punishment (see Chapter 3 ). Thorndike proposed that transfer occurs only to the extent that the original and transfer tasks ce. u d o r p ot re n o d , only The extensive training in estimating the areas oferectangles. training people’s subsequent ability to 3 improved s 0 u u 9 l d 0 a e . n 3 u o c triangles and circles) but had less estimate the areas of rectangles forms 202 o36 5.n(e.g., Pers and other two-dimensional @ 337 impact on judgments for the nonrectangular because nonrectangles have elements 1shapes—presumably o n i n . m both similar and dissimilar to those of rectangles. In a later study, Thorndike (1924) examined the have identical elements—that is, the two tasks involve some of the same specific stimulus–response associations. In an early study supporting his view (Thorndike & Woodworth, 1901), people received interrelationships of high school students’ achievement in various academic disciplines. Achievement in one subject area was correlated with students’ achievement in another only when the two subjects were similar. For example, arithmetic achievement was correlated with performance in a bookkeeping course, but Latin proficiency wasn’t. Thorndike concluded that the value of studying specific topics was due not to the benefits of mental exercise but to “the special information, habits, interests, attitudes, and ideals which they demonstrably produce” (Thorndike, 1924, p. 98). ce. u d o r p ot re n o A Later Behaviorist Perspective: Similarity of Stimuli and Responses d , y onl 9-03 e s u u is affected by l dtransfer 0 focused a e -have . n Subsequent to Thorndike’s work, behaviorist views of transfer onuhow 3 o 2 c s r 0 n . e 2 5 P 36situations. o @ stimulus and response characteristics in both the original and transfer As an illustration, consider 7 3 3 1 o n i these four lists of paired associates: m.n List 1 List 2 List 3 List 4 lamp–shoe lamp–sock rain–shoe lamp–goat boat–fork boat–spoon bear–fork boat–shop wall–lawn corn–road uce. d o r p e wall–yard tr nosofa–lawn o d , y l on e s -03 book–road u 9 l du 0 a e . n corn–lane 3 u o 2 c s r 0 n . e 2 P 365 o @ 7 3 13 m.nino wall–rice corn–fish Imagine that you first learn List 1 and then must learn List 2. Will your knowledge of the pairs in List 1 help you learn the pairs in List 2? Based on the results of verbal learning studies (J. F. Hall, 1966, 1971), the answer is yes: Stimuli are identical and responses are similar in the two situations, so positive transfer will occur. Now imagine that you have to learn List 1 and then List 3. Will prior learning of List 1 help with List 3? The answer again is yes (J. F. Hall, 1966, 1971). Even though the stimulus words are very different, the ce. u d o r p re been learned—and merely need ot already n responses in List 3 are identical to those in List 1—hence, they’ve o d , only 9-03 e to be attached to the new stimuli. s u u l a 2023-0o365.ncu.ed Person 7@ But now let’s suppose that after learning List 1, 3 you need to learn List 4. Here’s a case in which very different 3 1 o n i n . m be learned to the very same stimuli. Learning List 1 is likely to make responses from those of List 1 must learning List 4 more difficult because you’ll sometimes recall the List 1 response instead of the List 4 response (J. F. Hall, 1966, 1971), and negative transfer will result. In general, a stimulus–response view of transfer has yielded three central principles (Osgood, 1949; Thyne, 1963): ce. u d o r p ot re n o d , ly transfer When stimuli are similar and responses are different, negative will occur. on e 3 s 0 u u 9 l 0 a 2023- o365.ncu.ed Person @found that certain responses you make in As an example of the first and second principles, you 3 may 7have 3 1 o n .ni“pasting” in a word-processing program—also work in other mand one computer application—say, “copying” When stimuli and responses are similar in the two situations, maximal positive transfer will occur. When stimuli are different and responses are similar, some positive transfer will occur. computer applications—say, in e-mail or spreadsheets. As an example of the third principle, I recall a year when, as a high school student, my class schedule included second-period Latin and third-period French. The word for “and” (et) is spelled the same in both languages but pronounced very differently (“et” in Latin and “ay” in French), hence meeting the conditions for negative transfer (similar stimuli, different responses). On several occasions I blurted out an “et” in my third-period French class—a response that invariably evoked a disgusted scowl from my teacher. ce. u d o r p re otperspectives As learning theorists have moved away from behaviorist to more cognitive explanations of n o d , y l n e oof specific transfer, they’ve spoken lessl and Nevertheless, they agree sless -03 cu.edconnections. u u 9stimulus–response 0 a n 3 o 2 s r 0 n . 2 considerably 5 similarities exist between things already learned that the likelihood Peof transfer increases 36when o @ 7 3 13 and the demands of a new situation. m.nino An Information Processing Perspective: Importance of Retrieval From an information processing perspective, transfer can occur only when learners retrieve things they’ve previously learned at a time when those things might be useful (e.g., Cormier, 1987; Haskell, 2001; S. M. Lane, Matthews, Sallas, Prattini, & Sun, 2008; Rawson & Middleton, 2009). To make a connection between their current situation and any potentially relevant prior knowledge, learners must have the two things in working memory simultaneously. Given the low probability that any particular piece of information uce. d o r p e will be retrieved, as well as the limited capacity of working memory, notargood deal of potentially relevant o d , y l information and skills may very well not be transferred on to situations in which they would be helpful. e s u l a Person 9 03 d al Person -03 cu.edu 9 0 3 2 20to working 65.n A new event is more likely to call to mind information and ideas, if any, that are retrieved memory. 3 o @ 7 13of3the event and the needed information are closely associated o previously learned information when aspects n i n . m in long-term memory. For instance, this might be the case if learners previously anticipated the transfer The presence or absence of retrieval cues in the transfer situation influences the kinds of relevant situation when first storing the information. A Contextual Perspective: Situated Learning As you discovered in Chapter 11 , some cognitive theorists have proposed that a good deal of what we ce. u d o r p ot re n o d , situated learning is unlikely to result in transfer to very different (e.g., J. S. Brown, Collins, & only contexts e 3 s 0 u 9 l 0 .ed&uLester, 2009; Light na Collins, Duguid, 1989; Day & Goldstone,e 2012; 1996; Gresalfi u c rsoGreeno, n . 202&3Resnick, 5 P o36 @ & Butterworth, 1993). 7 3 3 1 m.nino learn is context specific—that is, it’s associated primarily with the environments and experiences in which learning has taken place, including the ways in which we have interacted with those environments. Such Even at school, knowledge and skills don’t necessarily transfer from one classroom to another. A study by Saljo and Wyndham (1992) provides an illustration. High school students were asked to determine how much postage they should put on an envelope weighing a particular amount, and they were given a table of postage rates with the information they needed. When students were given the task in a social studies class, most of them used the postage table to find the answer. But when students were given the task in a math class, most of them ignored the table and tried to calculate the postage in some manner, sometimes figuring it to several decimal places. Thus, the students in the social studies class were more likely to solve the ce. u d o r p ot reI suspect that they were well accustomed to problem correctly; as a former social studies teacher myself, n o d , onlyin that9context. looking for information in tables and charts contrast,umany of the students in math class e s -03 Incu u l d 0 a .eand n 3 o 2 s r 0 n . 2 that class drew on strategies (using formulas performing calculations) and so 5 Pe they associated with 6 3 o @ 7 overlooked the more efficient and accurate 133 approach. m.nino It’s important to note that not all cognitive theorists believe that school learning is as situated as some of their colleagues claim (e.g., see J. R. Anderson, Reder, & Simon, 1996, 1997; Bereiter, 1997; Perkins, 1992). For example, most individuals in our society engage in reading and simple mathematics—two sets of skills they’ve probably learned at school—in many nonschool contexts. A key factor here appears to be the extent to which learners perceive a content domain to be applicable to a wide range of circumstances (R. A. Engle, 2006; M. C. Linn, 2008; Mayer & Wittrock, 1996). For example, when baking cookies, an 11-year- ce. u d o r p ot re n o d , only to apply e 3 s u u 9-0mathematical l applicability, perhaps becauseo teachers ask students to many different 0 a .edprinciples n 3 u 2 c s r 0 n . e 2 5 P 36other disciplines, however. For example, situations and problems. The same isn’t necessarily@ trueofor 7 3 3 1 o they learn in algebra to a physics class, they rarely transfer although college students oftenm transfer .ninskills old might ask a parent, “Do two one-quarters make two fourths? I know it does in math but what about in cooking?” (Pugh & Bergin, 2005, p. 16). Fortunately, most students eventually learn that math has wide skills in the other direction, from physics to algebra (Bassok, 1997). A Contemporary View of General Transfer: Learning How to Learn We’ve already seen two extreme perspectives regarding general transfer. Advocates of formal discipline argued that learning rigorous and demanding subject matter facilitates virtually all future learning tasks because it “disciplines” and thus strengthens the mind. At the other extreme, Thorndike argued that learning duce e. c u d o r rep General transfer isn’t as tbetween: o n elements. Current perspectives about general transfer are somewhere in o d y, can facilitate nltime oone e s common as specific transfer, but learning occurring at -03 clearning u 9 l 0 a eduat another time if, in . n 3 u o 2 s r 0 n . e 2 5 P how to learn. the process, the individual learns o36 @ 7 3 3 1 inoHarlow m.nHarry In several early studies of learning to learn, (1949, 1950, 1959) found that monkeys and in one situation will transfer to another situation only to the extent that the two situations have identical young children became progressively faster at discrimination-learning tasks—that is, they became more efficient at learning which one of two or more objects would consistently lead to reinforcement. More recently, a number of studies have examined the transfer value of metacognitive knowledge and skills. Effective study strategies and habits, such as those described in Chapter 12 , often do generalize from one subject matter to another (S. M. Barnett & Ceci, 2002; Cormier & Hagman, 1987a; De Corte, 2003). ce. u d o r p t re oexamined n o d , Despite their differences, all the views ofntransfer we’ve so far have one thing in common: As long ly o e 3 s 0 u 9- norcuskills l u overlap02in3the-0information as two tasks have atsleast if the skills in question .edrequired—even r onasome . e 2 5 P 6 3 o are general metacognitive ones rather 3 than topic-specific ones—the possibility of transfer from one situation @ 7 3 1 o to the other exists. Yet m transfer .ninisn’t necessarily limited to cognitive and metacognitive acquisitions. For Going beyond Transfer of Knowledge: Emotional Reactions, Motives, and Attitudes May Transfer as Well instance, recall how, in the discussion of classical conditioning in Chapter 3 , Little Albert generalized (transferred) his fear of a white rat to other fuzzy white things. Just as emotional reactions can transfer to new situations, so, too, can motivation transfer. For instance, the kinds of goals students learn to set for themselves in a learning task—perhaps to truly master the subject matter, on the one hand, or to rotememorize just enough to “get by” on a class assignment or quiz—are apt to carry over as students move from one classroom to a very different one (Pugh & Bergin, 2005, 2006; Pugh, Linnenbrink, Kelly, ce. u d o r p ot re n o d , ononlylater9learning e s diverse viewpoints—can have a profound impact achievement -03 and u l 0 a .edu across multiple domains n 3 u o 2 c s r 0 n . e 2 5 P 6 De Corte, 2003; Pugh & Bergin, 2006; D. and so clearly illustrate general transfer at work (Cornoldi, o32010; @ 7 3 3 1 students develop a general desire to apply what they learn L. Schwartz, Bransford, & Sears,.n inoSome m 2005). Manzey, & Stewart, 2006; Volet, 1999). Furthermore, general attitudes and beliefs related to learning and thinking—for instance, recognition that learning often requires hard work, as well as open-mindedness to in the classroom—they have a spirit of transfer—that consistently resurfaces in later instructional contexts (Haskell, 2001; Pugh & Bergin, 2005, 2006). Factors Affecting Transfer Naturally, learners are more likely to transfer something they learn when they approach a learning task with ce. u d o r p ot re n o d , ly onlearning. e 3 chapters s u 9In-0previous Meaningful learning promotes better transfer than rote l 0 a .edu we’ve noted that n 3 u o 2 c s r 0 n . e 2 5 P 36already knows—leads to more meaningful learning—connecting new information with things oone @ 7 3 3 1 ino than does rote learning. Now we see an additional effective long-term memory storage and.n m retrieval a conscious intention to apply it. Several other factors also influence the probability of transfer, as reflected in the following principles. advantage of meaningful learning: It increases the odds of positive transfer (Brooks & Dansereau, 1987; Mayer & Wittrock, 1996; Schwamborn, Mayer, Thillmann, Leopold, & Leutner, 2010). For example, in one experiment (Mayer & Greeno, 1972), college students received one of two methods of instruction about a particular formula useful in calculating probabilities. Group 1 received instruction that focused on the formula itself, whereas Group 2 received instruction that emphasized how the formula was consistent with students’ general knowledge. Group 1 students were better able to apply the formula to problems e. c u d o r t rep o n o d nly, -03 o e s u l wider variety of situations. a -09 5.ncu.edu n 3 o 2 s r 0 e 2 6 likely it is to be transferred to a new situation. The more P thoroughly something is learned, the3more o @ 7 3 3 ino1The probability of transfer increases when students know something well Research is clear on .npoint: mthis similar to those they’d studied during instruction, but Group 2 students were better able to use the formula in ways instruction hadn’t specifically addressed—that is, they could transfer the formula to a (J. M. Alexander, Johnson, Scott, & Meyer, 2008; Cormier & Hagman, 1987a; Haskell, 2001; Voss, 1987). Thoroughly mastering knowledge and skills takes time, of course. In fact, some conditions that make initial learning slower and more difficult may actually be beneficial both for retention (see Chapter 8 ) and for transfer over the long run. For example, increasing the variability of tasks learners practice during instruction—having them perform several different tasks or several variations on the same task within a e. c u d o r single instructional unit—lowers their performance initially but tenhances rep their ability to transfer what o n o d they’ve learned to new situations (Z. Chen,o1999; & Bjork, 2008b; R. A. Schmidt & Bjork, nly,Kornell e 3 s 0 u 9 l 0 a edu . n 3 1992; van Merriënboer & Kester, 2008). u o 2 c s r 0 n . e 2 5 P o36 @ 7 3 3 Clearly, then, there’s a trade-off n ino1expediency and transfer. Teachers who teach a few things in m. between depth are more likely to promote transfer than those who teach many things quickly—the less-is-more principle introduced in Chapter 9 . The more similar two situations are, the more likely it is that something learned in one situation will be applied to the other situation. Behaviorists have argued that similarity of either stimuli or responses is necessary for transfer to occur. Cognitive theorists have instead proposed that transfer depends on retrieval of relevant information at the appropriate time, and thus the perceived similarity (rather than e. c u d o r t rep o n o d , nlyGeneral ofacts. Principles are more easily transferred than discrete principles and rules are more e 3 s 0 u 9 l 0 a ed&uHolyoak, 1987; . n 3 u o 2 c s applicable than specific facts and information (S. M. Barnett & Ceci, 2002; Gick r 0 n . e 2 5 P 6 o3example, @ 7 3 Judd, 1932; M. Perry, 1991; Singley & Anderson, 1989). For if you’ve read Chapter 3 , 3 o1 n i n . m you can probably recall the essence of operant conditioning: A response that is followed by a reinforcer is actual similarity) of the two sets of circumstances is important (Bassok & Holyoak, 1993; Day & Goldstone, 2012; Haskell, 2001; Voss, 1987). Either way, similarity enhances the probability of transfer. strengthened and therefore more likely to occur again. This principle is easily transferable to a wide variety of situations, whereas specific facts presented in the same chapter (e.g., who did what research study and when) are not. Similarly, when students are trying to understand such current events as revolutions and international wars, general principles of history—for example, the principle that two groups of people often engage in battle when other attempts at reaching a mutually satisfying state of affairs have failed—are probably more applicable than precise knowledge of World War II battles. e. c u d o r p General and perhaps somewhat abstract principles are helpful when a new situation does not, re ot especially n o d , on the surface, appear to be similarotonprevious and yet shares underlying structural or ly experiences e 3 s 0 u 9 l du the new situation requires far a those 2experiences—in conceptual similarities cu.ewhen 023-0o36other .nwords, ersonwith 5 P @& VanLehn, 2012; Perkins, 1995).2 transfer (J. R. Anderson et al., 1996; Chi 7 3 3 1 o n m.ni 2 Perkins and Salomon (1989, 2012) have made a distinction between low-road transfer and high-road transfer. Low- road transfer occurs when a new situation is superficially quite similar to prior experiences, such that relevant knowledge and skills are readily retrieved. But in high-road transfer, people must make a conscious and deliberate connection between a new situation and their previous experiences; such a connection is often made on the basis of an underlying, abstract principle. In other words, low-road transfer is most likely to occur in situations involving near uce. d o r p e notanrability to apply general principles to o d , y l As students move through the grade levels,othey may acquire n e s -03For example, u 9 l 0 a edinuone research study . n topics quite different from those they’ve previously studied. 3 u o 2 c s r 0 n . e 2 5 P o36students were asked to develop a plan for @ (Bransford & Schwartz, 1999), fifth-graders and college 7 3 3 ino1an endangered species in their state. None of the students in .neagles, increasing the population ofm bald transfer, whereas high-road transfer is required in situations involving far transfer. either age-group had previously studied strategies for eagle preservation, and the plans that both groups developed were largely inadequate. Yet in the process of developing their plans, the college students addressed more sophisticated questions than the fifth-graders did. In particular, the fifth-graders focused on the eagles themselves (e.g., How big are they? What do they eat?), whereas the college students looked at the larger picture (e.g., What type of ecosystem supports eagles? What about predators of eagles and eagle babies?) (Bransford & Schwartz, 1999, p. 67). Thus, the college students were e. c u d o r t rep o n o d ly, onpractice e s -03 the u 9 l Numerous and varied examples and opportunities for increase extent 0 a edtou which information . n 3 u o 2 c s r 0 n . e 2 5 and skills will be applied to P new situations. On average, the more o36examples and practice situations in @ 7 3 3 ino1 the greater the likelihood that future transfer will which particular information and skills .nencountered, mare drawing on an important principle they had acquired in their many years of science study: Living creatures are more likely to survive and thrive when their habitat supports rather than threatens them. occur (J. R. Anderson et al., 1996; Cormier & Hagman, 1987b; Cox, 1997; Perkins, 1995; R. A. Schmidt & Bjork, 1992). For instance, when students are learning basic arithmetic principles, they might be asked to apply those principles in such situations as determining best buys at a grocery store, dividing items equitably among a group of friends, and choosing inexpensive but healthful meals at a fast-food restaurant. Arithmetic will then be associated in long-term memory with all of these situations, and when the need arises to determine which of two grocery products yields the most for the money, relevant arithmetic procedures should be readily retrieved. e. c u d o r p the original task and the transfer task t rebetween o n The probability of transfer decreases as thed time interval o , onlyGick e increases (S. M. Barnettl&uCeci, 2002; Here’s s 03 u1987). u another principle that is probably 9&-Holyoak, d 0 a e . n 3 o 2 c s r 0 e Information that 2 has beeno3learned a result of P retrieval: 65.nrecently is more accessible and so more likely to @ 7 133 further back in time (recall the discussion of forgetting in be retrieved than information .ninoacquired m Chapter 8 ). Transfer increases when the cultural environment encourages and expects transfer. One of the most important factors influencing the probability of transfer is the general sociocultural context in which learning occurs (R. A. Engle, 2006; Haskell, 2001; Pea, 1987; Rogoff, 2003). In particular, when experienced adults and other individuals in a learner’s social environment communicate the importance of transfer and regularly point out similarities among seemingly diverse tasks and problems, the learner e. duc o r p e r not is more likely to apply relevant knowledge and skills to novel situations. Such a culture often exists in the workplace: New employees are expected to use newly acquired declarative and procedural knowledge in do , y l n o l use a n o s r Pe 9 03 a wide variety of work situations (Haskell, 2001; Wenger, 1998). It’s also evident in some classrooms, d a Person -03 cu.edu 9 0 3 2 n 20 for mysterious 65.purposes encouraged to acquire school subject matter (e.g., “You’ll need to know this in 3 o @ 7 3 3 college” or “It will come in handy noin1life”) but given little insight about when and in what ways it will .nilater m be useful. but probably isn’t as pervasive in schools as it should be. All too often, it seems, students are uce. d o r p e not r o d , y l e on -09-03 s u l du a e . n 3 u o 2 c s r 0 n . e 2 P o365 @ 7 3 3 1 m.nino rodu p e r t o do n , y l n o use 23-09-03 cu.edu l a n o s 20 Per 65.n 3 o @ 7 133 m.nino ce. . duce o r p e r not o d , y l e on -09-03 s u l a edu . n 3 u o 2 c s r 0 n . e 2 5 P o36 @ 7 3 3 1 m.nino duce e. oduc r p e r t no o d , y l n se o 23-09-03 u.edu u l a n o 20 5.nc Pers 6 3 o @ 1337 o n i n . m uce. d o r p e not r o d , y l on e s 03 u.edu u 9 l 0 a n 3 o 2 .nc 20 5 Pers 6 3 o 37@ 3 1 o n i m.n ce. odu r p e r t do no , y l n o use 23-09-03 cu.edu l a n o s 20 Per 65.n 3 o @ 7 133 o n i n . m e. c u d o r t rep o n o d only, 9-03 e s u l a -0 5.ncu.edu n 3 o 2 s r 0 e 2 P o36 @ 7 3 3 1 m.nino e. oduc r p e r t no o d , y l n se o 23-09-03 u.edu u l a n o 20 5.nc Pers 6 3 o @ 1337 o n i n . m e. oduc r p e r t no o d , y l n se o 3-09-03 u.edu u l a n o 202 o365.nc Pers 37@ 3 1 o n i m.n Problem Solving ce. u d o r p ot re question or troubling situation. Following are n o d learned—that is, transferring them—to address an unanswered , only 09-03 e s u u l examples: a 2023- o365.ncu.ed Person 7@ 3 3 1 1. What number is obtained when 3,354 is divided by 43? o n ni . m 2. How can a 60-something educational psychologist be helped to control her junk food habit? When we talk about problem solving, we’re talking about using knowledge and skills we’ve previously 3. How can two groups of people of differing political or religious persuasions and a mutual lack of trust be convinced to curtail their proliferation of military weapons and work toward cooperation and peaceful coexistence? The world presents us with many different kinds of problems. Some, such as Problem 1, are straightforward: All the necessary information is presented, and the solution is definitely right or wrong. Others, such as ce. u d o r p ot re n o d , y onl 09-03 e s u l dtouthe problem (e.g., selfa -two or more e . n 3 with Friends on her smartphone?); there may2also be solutions u o 2 c s 0 r n . e P 365island with no mail deliveries or Internet o @ reinforcement for altered eating habits, six months on a remote 7 3 13 access). Still others, such as Problem m.ni3,nomay be so complicated that even after considerable research and Problem 2, may necessitate seeking out additional information (e.g., does the educational psychologist keep physically active, or does she sit around the house all day reading professional journals and playing Words creative thought, no easy solution emerges. Different kinds of problems require different procedures and different solutions; this multifaceted nature of problem solving has made the theoretical study of problem solving a very challenging endeavor indeed. Any problem has at least three components (Glass, Holyoak, & Santa, 1979; Wickelgren, 1974): e. c u d o r t rep o n o d Operations. Actions that can be performed to approach the goal onlory,reach e 03 u.edu s u 9 l 0 a n 3 o 2 s 0 nc 2 .then 5 Per 6 3 o Operations often take the form of IF–THEN rules: If I get such-and-such, I next need to do such-and@ 7 3 3 1 o in 9 ). such (recall the discussion of productions min.nChapter Goal. The desired end state—what a problem solution will hopefully accomplish Givens. Pieces of information provided when the problem is presented Problems vary considerably in terms of how well the three components are specific and clear. At one end of the continuum is the well-defined problem, in which the desired end result is clearly stated, all needed information is readily available, and a particular sequence of operations will (if properly executed) lead to a correct solution. At the other end is the ill-defined problem, in which the goal is ambiguous, some essential information is missing, and there’s no guaranteed means of achieving the goal. Well-defined problems often have only one right solution, whereas ill-defined problems often have several possible solutions that vary in ce. u d o r p terms of their relative rightness and acceptability. The problem ot reof dividing 3,354 by 43 is well defined, while n o d , ly the goal that of military disarmament is ill defined (e.g., of “cooperation and peaceful coexistence” is on 3 e 0 s u u to solve and require more 9 l 0 3 a .edharder u ambiguous). As e are typically on guess, ill-defined c n 2023problems . 5 P yoursmight o36 problems. @ sophisticated problem-solving strategies than well-defined 7 3 3 1 m nino 3 m.ni For an in-depth discussion of ill-defined problems in international relations, see Voss, Wolfe, Lawrence, and Engle (1991). Unfortunately, researchers have focused more on well-defined problems—often somewhat artificial ones— than on the ill-defined problems that life so often presents. You’ll probably notice this bias as we proceed with our discussion of problem solving. Nevertheless, most of the theories and principles I’ll present in the upcoming pages presumably apply to both kinds of problems. And in fact, later in the chapter we’ll find that e. duc o r p e r not o d , y l e on 3-09-03 s u u l a n 202 o365.ncu.ed erso P Theories of Problem Solving 37@ 3 1 o n i n m. one strategy for solving ill-defined problems is to pin them down a bit—in other words, to translate them into well-defined ones. Behaviorists and cognitivists alike have offered theories of how human beings and other animal species solve problems. Here we’ll look briefly at early behaviorist and cognitivist views of problem solving. We’ll then draw largely from information processing theory as we explore more contemporary explanations. Early Behaviorist Views: Trial-and-Error Learning and Response Hierarchies In Chapter 3 , you read about Thorndike’s (1898) classic work with a cat in a puzzle box. In trying to solve ce. u d o r p ot re n o d , y nl mechanism. once again tried various behaviors until it triggered the With successive trials, the cat orelease 3 e 0 s u 9 l du its general 0 a e . n 3 u o 2 c s managed to escape more quickly—and hence was rewarded for the correct response—but 0 r n 2 . e P 365 o @ 7 approach was largely one of trial and error. 3 13 m.nino the problem (getting out of a confining situation), the cat explored the box, manipulating its various parts and eventually triggering a mechanism that opened the door. The poor cat was returned to the box, where it A trial-and-error approach is often observed in children’s problem-solving behavior. For example, consider the way that many young children assemble jigsaw puzzles: They try to fit different pieces in the same spot, often without considering each piece’s shape and coloring, until eventually they find a piece that fits. Such a strategy is workable only if the number of possible solutions is quite small. In trial-and-error learning, humans and nonhumans alike may discover that they can potentially solve a particular problem in a variety of ways, with some ways having greater success rates—that is, they’re uce. d o r p e not r o d , y l on 09-03 e s u l du 1966). Such a response hierarchy a – 2009; e . n 3 1966; Hull, 1938; Shabani, Carr, & 2 Petursdottir, B. F.u Skinner, o 2 c s 0 r n . e 5 stronger associations): P o36indicate can be graphically depicted like this (thicker arrows @ 7 3 3 1 m.nino reinforced more often—than others. The result is that the stimuli comprising the problem might be associated with several responses, with some associations being stronger than others (e.g., G. A. Davis, As an example, when my daughter Tina was growing up, she often had the same problem: getting permission to do something her father and I didn’t want her to do. Given this problem, she typically tried three different responses, usually in the same order. First, she’d smile sweetly and describe how much she . d duce o r p e r not rep t o n o ,d 3 nlyunproductive, o e s this particular problem situation). a If that tactic was speak u l eduindignantly about how her -09-05.nshe’d . n 3 u o 2 c s r 0 e 2 6off to her room, slamming her door and parents never let herP do anything. As a last resort, she’d run 3 o @ 7 3 13 did learn that some problems would be solved only over her shouting that her parents hated her. inonever .nTina m parents’ dead bodies. wanted to engage in the forbidden activity (such a response, apparently, was associated most strongly with Although trial-and-error learning and response hierarchies are sometimes helpful in understanding problemsolving behavior, contemporary theorists have largely abandoned behaviorist approaches to focus more on cognitive processes involved in problem solving. Consistent with this trend, we, too, will abandon behaviorism at this point and embrace a more cognitively oriented perspective for the rest of our discussion. ce. u d o r p ot re n o d , In Chapter 6 , I described Gestalt psychologist Wolfgang (1925) observations of chimpanzee onlyKöhler’s 3 e 0 s u 9 l 0 a .edu of the form n 3 behavior in problem-solving situations. Köhler observed very little trial-and-error behavior u o 2 c s 0 r n 2 . e 5 P 6 carefully examining the o3were @ Thorndike had described. Rather, it appeared that the3 chimpanzees 7 3 ninoup,1 so to speak—and mentally configuring and components of a problem situation—sizing m.things Early Cognitivist Views: Insight and Stages of Problem Solving reconfiguring those components until they found a winning combination. At this point of insight, the chimps would immediately spring into action, deliberately making a sequence of responses to solve the problem. In Köhler’s view, then, problem solving is a process of mentally restructuring a problem situation until insight is achieved. Another early cognitive approach to problem solving was to identify the mental stages— perhaps including ce. u d o r p ot re n o d , only 09-03 e s u l du 1. Preparation. s Defining the problem relevant information a 3-gathering u.e on c 02and r n 2 . e 5 P 6 oa3subconscious level while engaging in other activities 2. Incubation. Thinking about the problem at @ 7 3 3 1 ino insight into a problem solution 3. Inspiration. Having m.ansudden insight—through which problem solving might proceed. For example, Wallas (1926) identified four steps in problem solving: 4. Verification. Checking to be certain that the solution is correct Whereas Wallas suggested that certain aspects of problem solving might be out of the mental limelight, Polya (1957) proposed four steps that rely heavily on conscious, controlled mental activities: 1. Understanding the problem. Identifying the problem’s knowns (givens) and unknowns and, if appropriate, using suitable notation (e.g., mathematical symbols) to represent the problem uce. d o r p e ot r n o d , y onl 09-03 e s u l duthe intention of learning a e . n 3 4. Looking backward. Evaluating the overall effectiveness of.the plan, with u o 2 c s 0 r n 2 e 5 P 36future something about how to solve similar problems inothe @ 7 3 13 m.nino 2. Devising a plan. Determining appropriate actions to take to solve the problem 3. Carrying out the plan. Executing the plan and monitoring its effectiveness Unfortunately, Wallas and Polya derived their portrayals of problem solving more from introspection and informal observation than from controlled experimentation, and they were vague about how a learner might accomplish each step (Lester, 1985; Mayer, 1992; Schoenfeld, 1992). For such reasons, these early stage theories of problem solving have been of only limited usefulness in facilitating people’s problem-solving success. uce. d o r p e ot r n o d , y onl 09-03 e s u u Wallas l dwhich a – processes e . n 3 those Polya described) and at other times involves less conscious (about u o 2 c s 0 r n 2 . e P 365 problems with which they’re o @ speculated). Furthermore, learners sometimes reconceptualize (restructure) 7 3 13 of the sudden-insight variety. inosolutions wrestling and, in doing so, can occasionally m.nreach Yet early cognitive views have clearly had an impact on more recent explanations of problem solving. As we’ll soon see, problem solving sometimes involves deliberate and controlled mental processes (similar to Information Processing Theory Most contemporary researchers take an information processing approach and focus largely on specific cognitive processes that contribute to problem-solving success. In the following section we’ll examine such processes. . e Cognitive Factors Affecting Successful Problem Solving produc t re o n o d y, on onldepends The ability to solve problems successfully of the human information processing 3several aspects e 0 s u u 9 l d 0 a e . n 3 rso capacity, encoding ncu memory retrieval, existing knowledge system: working 202 processes, .long-term 5 Pememory 6 3 o 37@ relevant to the problem, and metacognition. As we examine these factors, we’ll identify several differences 3 1 o n i n . m between problem-solving experts and novices within a particular content domain, discovering reasons why some people solve problems easily and effectively, whereas others solve them either with great difficulty or not at all. Working Memory Capacity As you should recall, working memory is the component of memory in which active, conscious thinking ce. u d o r p t re of working memory’s capacity—the ntooomuch o d working memory—or perhaps if irrelevant thoughtsly consume , o&nAlloway, e 03 Hambrick s u 9-2006; l problem can’t be solved (Gathercole, Lamont, 0 a .ed&uEngle, 2003; Wiley & n 3 u o 2 c s 0 r n 2 . e 5 P o36 Jarosz, 2012). @ 7 3 3 ino1 n . m Problem solvers can overcome this working memory limitation in a couple of ways. First, some of the occurs. Yet this component can hold and process only a small amount of information at a time. If the information and mental processes necessary to solve a problem impose too much of a cognitive load on information necessary to solve the problem can be stored externally (e.g., by writing it on paper or putting it on a computer screen) or perhaps even processed externally (e.g., by using a calculator). Second, as suggested in Chapter 8 , some skills involved in problem solving should be learned well enough that they become automatic, thus leaving most of working memory capacity available for more challenging aspects of the problem. uce. d o r p e not r o d , Consider this classic children’s riddle: y l on 09-03 e s u l du a e . n 3 u o 2 c s 0 r n 2 . e As I was going to St. Ives,P 365 o @ 7 3 13 I met a man with seven wives. m.nino Encoding of the Problem Every wife had seven sacks. Every sack had seven cats. Every cat had seven kits. Kits, cats, sacks, wives. How many were going to St. Ives? ce. u d o r p e to St. Ives. People who solve the problem ot rgoing plus 7 cats (343) plus 7 kits (2,401) equal a,total ofn 2,802 o d y l oInnparticular, 3 overlooked e s -0they’ve u 9 in this way have encoded it incorrectly. l du the first line of the riddle: “As I was 0 a e . n 3 u o 2 c s 0 r n 2 . e 5 where the polygamist was taking his wives and P The problem statement doesn’t 3tell6us going to St. Ives.” o @ 7 3 o13 not. menagerie—maybe to St..Ives, m ninmaybe Many people take this logical approach to the problem: 1 traveler plus 1 man plus 7 wives plus 72 sacks (49) 3 4 One critical factor in encoding a problem is determining what aspects of the problem are relevant and irrelevant to finding a solution. A second, related factor is how various aspects of the problem are encoded (K. Lee, Ng, & Ng, 2009; Mayer & Wittrock, 2006; Ormrod, 1979; Whitten & Graesser, 2003). For example, consider these two ways of presenting the same situation: There are 5 birds and 3 worms. How many more birds are there than worms? There are 5 birds and 3 worms. How many birds won’t get a worm? uce. d o r p e not r o d , y l 3 the second e onand First-graders often struggle with the first problem yet can one quite easily (Hudson, s -0solve u u 9 l d 0 a e . n 3 u o 2 c s 0 relational r students to2store 5.n Pe 1983). The first problem requires one thing compares to 6information—how 3 o @ 7 3 3 another. Relational information seemsin .n toobe1difficult to encode, even for adults (Mayer & Wittrock, 1996). m For example, in one study (Mayer, 1982), college students were asked to remember problems such as this one: A truck leaves Los Angeles en route to San Francisco at 1 P.M. A second truck leaves San Francisco at 2 P.M. en route to Los Angeles going along the same route. Assume the two cities are 465 miles apart and that the trucks meet at 6 P.M. If the second truck travels at 15 mph faster than the first truck, how fast ce. u d o r p ot reof problems (e.g., one truck n o d The students made three times as many errors in recalling relational aspects , onlyassertions e s u 9-03 l 0 traveling 15 mph faster than another) than in recalling basic (e.g., two a edubeing 465 miles .cities n 3 u o 2 c s 0 r n 2 . e 5 P o36 apart). @ 7 3 3 1 nino . m How people encode a problem—and therefore how they solve it—is partly a function of how they classify the does each truck go? (Mayer, 1982, p. 202) problem to begin with. For example, if you initially perceived the St. Ives poem to be a math problem, you probably did a lot more mental work than you needed to and ended up with an incorrect answer. People are apt to have a variety of problem schemas—knowledge about certain types of problems that can be solved in certain ways—that they use in problem classification (G. Cooper & Sweller, 1987; L. S. Fuchs et al., 2004; Mayer & Wittrock, 2006). For example, consider this problem: uce. d o r p e ot r 1989, p. 165) n(Resnick, o d left. How much did Ana have when she n started out? , y l o al use 9 03 d Ana went shopping. She spent $3.50 and then counted her money when she got home. She had $2.35 o y 3 e s u l a If you retrieve and apply problem, -0read the du you’ll probably get the correct answer e . 3-0as9you ersoann addition2schema u 2 P c 0 n . 65 you to classify the problem as one requiring of $5.85. However, if the word left in the problem o3leads @ 7 3 3 1 subtraction—and after all, problems ask questions such as “How many are left?”—you inosubtraction .nmany m may very well get the incorrect answer of $1.15 (Resnick, 1989). Problem classification comes into play when solving social problems as well. Consider this situation: The students in Alice’s ninth-grade social studies class have been working in pairs on an assigned project; their teacher has said that he’ll give a prize for the best project a pair completes. Alice is now complaining ce. u d o r p too bossy. Louisa suggests that the best course of action is simply ot refor Alice and Meg to buckle down and n o d , finish the project so they can win the prize. Alice, is3thinking that she should instead talk with onlyhowever, e 0 s u u 9 l 0 a u.ed1994) c 023onoBerg rsobenless bossy.2(Based n Meg and promise that she’ll & .Calderone, 5 Pe 6 3 @ 7 3 3 1 ino in different ways and thus arrive at different solutions. In Louisa and Alice are classifyingm the.n problem to her younger sister Louisa that her partner, Meg, no longer wants to work with her; Meg thinks Alice is Louisa’s eyes, the problem is one of completing the project. Alice sees the problem quite differently—as one of resolving an interpersonal conflict (Berg & Calderone, 1994). Experts and novices in a particular content domain tend to classify problems differently (Anzai, 1991; M. T. H. Chi & VanLehn, 2012; De Corte, Greer, & Verschaffel, 1996). Experts generally classify a problem on the basis of abstract concepts and underlying principles and patterns. They seem to have a well-developed set of problem schemas they use to represent different kinds of problems. In contrast, novices tend to focus on uce. d o r p e specific, concrete aspects of a problem and so are apt to retrieve information not r related only to those aspects. o d , y l As an illustration, Schoenfeld and Herrmann (1982) ways professors and on 0the 3 that mathematics e compared 0 s u u 9 l d a e . n u on abstract principles students categorized a variety P of e math them rsoproblems. Professors 2023classified .ncbased 5 6 3 o 7@grouped together, as were those solved by related to problem solution: Problems solved by analogy 3were 3 1 o n i m.n classified the problems based on more superficial contradiction, and so on. In contrast, students characteristics, such as whether they contained polynomial expressions or included geometric diagrams. After taking a math course, the students repeated the classification task; at this point, they began to classify the problems as their professors had. Experts may also spend a good deal of time defining ill-defined problems before attempting to solve them (J. B. Mitchell, 1989; Swanson, O’Connor, & Cooney, 1990; Voss, Tyler, & Yengo, 1983; Voss, Wolfe, e. c u d o r t rep o n o d Imagine that you’re the Minister of Agriculture in a developing country. Crop productivity has been low for only, 09 3 e 0 s u l du would you do to increase crop a people2are – to5go.nhungry. eWhat . n 3 u o 2 c s the past several years, and beginning 0 r e P 6 o& 3Penner, @ 7 3 production? (Based on Voss, o Greene, Post, 1983; Voss, Tyler, et al., 1983) 3 1 m.nin Lawrence, & Engle, 1991). Consider this problem as an example: Take a few minutes to jot down some of your ideas. How much time did you spend defining the problem? Chances are, you didn’t spend much time at all; you probably went right to work thinking about possible problem solutions. But if you were a political scientist specializing in that country, you would probably have spent considerable time identifying various aspects of the problem—perhaps including local government policies and practices, the amount of land available for e. c u d o r t rep o n o d only, 9-03 e s u l a -0 5.ncu.edu n 3 o 2 s Mental sets in encoding r 0 e 2 P o36 @ 7 3 3 People are often predisposed to approach ino1and encode problems in particular ways—a phenomenon known n . m 4 farming, recent climatic conditions, and available technologies—before thinking about how you might solve it (J. B. Mitchell, 1989; Voss, Greene, et al., 1983; Voss, Tyler, et al., 1983). as mental set. Here’s a problem for which people are often the victims of mental set: 4 Gestalt psychologists introduced this idea, using the term Einstellung. How can you throw a tennis ball so that it goes a short distance, comes to a complete stop, and then reverses its direction? You may not bounce the ball against a surface, nor may you attach any other object ce. u d o r p t raehandful of my 35 students oonly n o d I once gave this problem to a masters-level learning theories class, and , nly 9-03 othat e s u u l could solve it. Most of them worked on o the assumption to be thrown (some a 3-0ball had u.edhorizontally c s n 02the r n 2 . e 5 P 6 o3Once you break this mental set, the even said they encoded the problem as a visual image of a pitcher). @ 7 3 3 1 answer’s quite simple: You throw the ball mup..nino (such as a string) to it. (Based on M. Gardner, 1978) As another example of mental set, consider this problem: You have a bulletin board firmly affixed to the wall. Your task is to stand a thin candle upright beside the bulletin board about 4 feet above the floor. You don’t want the candle touching the bulletin board (a fire hazard) but instead need to place it about a centimeter away. How can you accomplish the task with the uce. d o r p e tr noreading. o Develop a solution for the problem before lyou continue d , y on 09-03 e s u l du a – activity e . n 3 u o 2 c s 0 r n 2 . When I’ve presented this problem as a hands-on in graduate classes, students have typically e 5 P 6 3 o @ 7 is to stab the knitting needle through the candle and into the 33solution identified three different solutions. One 1 o n i n . malmost invariably splits the candle and gouges the bulletin board. A second bulletin board; this action materials shown in Figure 13.1 ? (Based on Duncker, 1945) solution is to form a horizontal surface with the ruler, propping it against the bulletin board with thumbtacks (possibly also using the knitting needle) and place the candle on top; however, the precariously placed ruler usually falls to the floor once the candle is added. Only a third solution works: Take the thumbtacks out of their box, attach the box to the bulletin board with tacks, then affix the candle to the top side of the box with either melted wax or a tack. The solution is obvious once you think about it, but many people encode the . duce o r p e r not o d , y l e on 3-09-03 s u l a edu . n u o 2 c s 0 r n 2 . e 5 P o36 @ 7 3 3 1 m.nino box only as a container and fail to consider its other possible functions. . duce o r p e r not o d , y l e on 3-09-03 s u l a edu . n u o 2 c s 0 r n 2 . e 5 P o36 @ 7 3 3 1 m.nino Figure 13.1 Using some or all of these materials, how can you stand a candle upright beside the bulletin board so that you can safely light it? ce. u d o r p ot re n o d , ly -experience onindividuals McCaffrey, 2012). The degree toswhich functional fixedness depends, in part, on 3 e 0 u 9 l 0 a .edifuthe tacks are presented outside the n 3 u c n situational conditions. candle problem 202solveothe . 5 PersoPeople more easily 36 @ 7 3 box, decreasing the likelihood that the box is encoded as a container (Duncker, 1945). The problem is also 3 1 no i n . m easier to solve if the box itself is labeled “BOX” (Glucksberg & Weisberg, 1966), possibly because the label The tendency to think of objects as having only one function, thereby overlooking other possible uses, is a form of mental set known as functional fixedness (e.g., Birch & Rabinowitz, 1951; Duncker, 1945; draws attention to the box as something that can be used in solving the problem. Mental set and functional fixedness are partly the result of past experience: If a particular approach to a problem has worked in the past, a person is likely to continue using it and possibly learn it to automaticity. The person is then likely to apply this approach—often in a relatively “mindless” fashion—even in situations where it’s inappropriate or unnecessarily cumbersome. (Recall the discussion of automaticity’s downsides in Chapter 8 .) ce. u d o r p ot re n o d , y Luchins’s classic experiments with water jar problems Luchins & Luchins, 1950) illustrate just how onl 0(1942; 3 e 0 s u u you have three jars of 9 l dthat a e . n 3 u o strongly past experiencesecan influence problem solving. Imagine, ifcyou will, 2 s 0 r n 2 . P 365 o @ three different sizes: 7 3 13 m.nino Jar A holds 20 ounces of water. Jar B holds 59 ounces of water. Jar C holds 4 ounces of water. You need exactly 31 ounces of water. Given an unlimited water supply, how could you get the exact amount of water using only the three jars? Try to find a solution before you read further. . duce o r p e r not o d , The solution to the water jar problem is as follows: y l e on 3-09-03 s u l a n du e . 202 u Perso c n 365 2 cu.e n . 5 6 3 @o 7 3 3 1 o 2. Pour water from Jar B into Jar A until A is full. This leaves 39 ounces in Jar B. .nin m 3. Pour water from Jar B into Jar C until C is full. This leaves 35 ounces in Jar B. Per 1. Fill Jar B. This gives you 59 ounces. 4. Pour the water out of Jar C and once again fill it with water from B. At this point you have 31 ounces in Jar B, the exact amount you need. Mathematically speaking, the solution to the problem is B − A − 2C. uce. d o r p e not r o d , y l e on 3-09-03 s u l a edu Obtain this amount: . n u 2 Jar A holds: erso Jar B holds: Jar C5holds: c 0 n 2 . P o36 @ 7 3 3 1 23 3 20 m49.nino Luchins (1942) gave people a series of such problems, with the answer always being the same: B − A − 2C. He then presented the following three problems: 15 39 3 18 28 76 3 25 Almost everyone solved the first two problems using the same formula as before: B − A − 2C. They had e. c u d o r t rep o n o d y, + C. The participants in Luchins’s study (who were and students—hardly mental onlprofessors 3 graduate e 0 s u u 9 l d 0 a e . n 3 u o 2 slouches!) were victims of ars mental set established through prior experience. c 0 n 2 . e 5 P o36 @ 7 3 3 no1 similar problems in similar ways facilitates successful In many situations a predisposition to iapproach m.n trouble solving the third problem because the standard formula didn’t apply. But notice that all three can be solved quite easily: The solution for the first and third problems is A – C, and that for the second problem is A problem solving. A mental set influences the way in which a problem is encoded in memory, however, and this encoding in turn influences the parts of long-term memory that are searched for potentially relevant information and procedures. If a person’s encoding of a problem steers that person in an unproductive “direction” in long-term memory, it’s apt to hinder problem-solving performance (Bilalic, McLeod, & Gobet, 2010; N. R. F. Maier, 1945; Mayer, 1992; D. L. Schwartz, Chase, & Bransford, 2012; Stein, 1989; R. J. Sternberg, 2010). ce. u d o r p ot re n o d , ly must onpeople To use previously acquired knowledge to solve a problem, they’re thinking e 03retrieveuit.ewhile s u u 9 l d 0 a n 3 o about the problem. Thus, storage that facilitate memory rs 202long-term 5.ncretrieval—for instance, Peprocesses 6 3 o @ meaningful learning and integration of new ideas—facilitate 1337problem-solving success as well. o n i n . m Retrieval from Long-Term Memory When people search long-term memory for knowledge relevant to a problem, they begin by looking in logical “places.” They tend to retrieve familiar ideas first, identifying original or unusual problem solutions—those that require thinking “outside the box”—only later on, if at all (Bourne, Dominowski, Loftus, & Healy, 1986; Storm & Angello, 2010). People also tend to retrieve information closely associated with aspects of the problem situation; for example, they can more easily solve the candle problem if they’ve previously learned a e ce. u d o r p re as well, provided that people perceive their obet helpful 1979). Hints that provide important retrieval cuesocan n d , nly & Holyoak, oGick relevance (Bassok & Holyoak, 1993; e s u 9-03 n1987). l 0 a .edu n 3 u o 2 c s 0 r 2 . e 5 P 36 during problem solving is anxiety. We’ll look at oprocesses @ One factor that can hinder productive 3 retrieval 7 3 1 .nininoChapter 14 , but for now you should note that anxiety can interfere with anxiety’s effects in more mdetail paired-associates list that includes the pair candle–box (Bassok, 2003; Weisberg, DiCamillo, & Phillips, retrieval by restricting the part of long-term memory being searched and thus reducing the chances of finding useful information (Ashcraft, 2002; Beilock, 2008; Zeidner & Matthews, 2005). For instance, imagine that you have a very simple problem: You can’t find your car keys. You’re desperate to find them quickly because you’re already late for an important appointment. You look in the same places over and over again; you don’t think creatively about the wide variety of places in which the keys might be lurking. uce. d o r p e ot r n o d , y nl similar college students were asked to solve a candle to that presented in Figure 13.1 . Some oproblem 3 e 0 s u 9 l du whereas others were 0 a e . n 3 u o 2 c s students were given the tacks outside of the box (an easy version of the problem), 0 r n 2 . e P 365 for each version of the problem, some o @ 7 given the tacks inside the box (a more difficult version). Furthermore, 3 ninupoto13$25 for solving the problem quickly—a tidy sum back in 1962, m.earn students were told that they could In general, anxious individuals are apt to have trouble solving problems whose solutions aren’t readily apparent (Glucksberg, 1962; B. Hoffman, 2010). For example, in one early study (Glucksberg, 1962), and one that would significantly increase students’ stress levels as they worked on the problem. Following are the average reaction times (in minutes) for the four groups of problem solvers; larger numbers indicate greater difficulty in solving the problem: Easy version Difficult version e. c u d o r t rep o n o d only, 09-11.08 High anxiety 3.67 e 03 u.edu s u l a n 3 o 2 s 0 2 .nc 5 Per 6 3 o 37@ 3 1 When the box was empty, its use as a platformin forothe candle was obvious, and anxiety facilitated problem n . m solving. But when the box was already being used as a container, anxiety made the problem much more Low anxiety 4.99 7.41 difficult to solve than it would have been otherwise. The effects of anxiety on problem solving seem to be reduced or eliminated when people know where to search in long-term memory. Although high-anxiety individuals typically perform more poorly than lowanxiety individuals on problem-solving tasks, performance of the two groups is similar when memory aids are provided to facilitate appropriate retrieval (Gross & Mastenbrook, 1980; Leherissey, O’Neil, & ce. u d o r p ot re n o d , only retrieval e The value of incubation in long-term memory s -03 cu.edu u 9 l 0 a n 3 o 2 20 5.nearlier, one key step is incubation: letting the Pers 6 3 o In Wallas’s four-step theory of problem solving presented @ 37 problem “percolate” form a while—perhaps .nino13 at an unconscious level—while engaging in other activities Hansen, 1971). (Wallas, 1926). Many contemporary cognitive theorists have also vouched for the importance of incubation, especially in dealing with difficult problems. For one thing, some of the factors that interfere with problem solving (e.g., fatigue, anxiety, counterproductive mental sets) may dissipate during the incubation period. And in the intervening time period, a person can search long-term memory more broadly—perhaps simply by “wandering aimlessly” through various areas (not necessarily with any conscious purpose or intent) and ce. u d o r p t re o n o d , and so tackle it differently. In some cases, the o recoding yield an almost instantaneous solution, resulting nly can e 03 u&.Wiley, s u 9 l 0 a edu2006; Baird et al., 2012; in that sudden insight phenomenon of which I spoke earlier (I. K. Ash n 3 o 2 c s 0 r n 2 . e 5 P 36 2010; Topolinski & Reber, 2010; Zhong, Kounios & Beeman, 2009; Strick, Dijksterhuis, &@ vanoBaaren, 7 3 3 1 Dijksterhuis, & Galinsky, 2008). m.nino unexpectedly “stumbling” on a potentially helpful idea. When encountering the idea, the person may see its relevance for the previously unsolved problem, encode the problem in a new way—that is, restructure it— In my own experiences in writing various editions of this book, I’ve found incubation to be an extremely useful strategy. Probably the biggest problem I face when I write is figuring out how best to organize the ever-expanding body of research findings related to learning and motivation—and also to keep this book to a reasonable size! As my field continues to grow and evolve, I find that some organizational structures I’ve used in previous editions of this book are no longer useful in a later edition, and so I begin to experiment ce. u d o r p ot re n o d , y perhaps watch a television game show or two, and e essentially dust” settle. When I return to onl let09the-0“mental 3 s u uthe day before. l d a e . n 3 u o my computer the following morning, I often have fresh ideas that hadn’t occurred to me 2 c s 0 r n 2 . Pe 365 o @ 7 3 13 Knowledge Base m.nino with alternative arrangements. Yet my mind seems to be able to handle only so much mini-paradigm-shifting in any single day. Oftentimes the best thing I can do is to turn off my computer in midafternoon, take a walk, On average, successful problem solvers have a more complete and better organized knowledge base for the problems they solve. And if they’re experts in their field, they also have more knowledge of specific problem-solving strategies within their area of expertise. For instance, once they’ve categorized a problem as falling into a particular category or being consistent with a certain problem schema, they readily apply certain procedures to solve it, and they’ve often learned basic problem-solving procedures to automaticity. Lacking the rich knowledge base that experts have, novice problem solvers are more likely to engage in e. c u d o r t rep o n o d ly, -03 onBédard e s u rote, meaningless manner (Anzai, 1991; M. l du 2010; Chi, Glaser, & Farr, 1988; a -09& Chi,5.1992; eCarr, . n 3 u o 2 c s 0 r n 2 e P Lawson & Chinnappan, 1994; S. K. Reed, 1993). o36 @ 7 3 3 1 m.nino Metacognition ineffective problem-solving strategies—for example, resorting to trial and error; persevering with unproductive procedures; making unwarranted assumptions; and applying procedures and equations in a Metacognition often plays key roles in problem solving. In particular, successful problem solvers must Believe that they have sufficient knowledge to solve a problem successfully Realize that some problems may take considerable time and effort to accomplish Plan a general course of action Flexibly consider potentially relevant problem-solving strategies and choose appropriate ones e. duc o r p e r not o d , y l n se o 23-09-03 u.edu u l a n o 20 5.nc Pers 6 3 o @ 37 nino13 Monitor progress toward a solution, and change strategies if necessary (M. Carr, 2010; Dominowski, 1998; B. Hoffman & Spatariu, 2008; Kirsh, 2009; Mayer & Wittrock, 2006; Zimmerman & Campillo, 2003) 133 m.nino Students sometimes have epistemic beliefs that interfere with effective problem solving. For instance, as noted in Chapter 12 , many students in mathematics classes believe that (a) math problems can have only one right answer, (b) there’s only one way to solve any particular problem, (c) problem-solving procedures don’t necessarily make sense, and (d) a problem is either solvable within a few minutes’ time or else not solvable at all (De Corte, Op’t Eynde, Depaepe, & Verschaffel, 2010; Muis & Foy, 2010; Richland, Stigler, & Holyoak, 2012). And when students have naive beliefs about the nature of a subject area or about knowledge more generally— perhaps thinking that virtually any issue has only a single correct solution e. c u d o r t rep o n o d only, 09-03 e s u l a edu . n 3 u o 2 c s 0 r n 2 . e 5 P Problem-Solving Strategies o36 @ 7 3 3 ino1 problem-solving strategies. Such strategies fall into mfor.nappropriate At several points I’ve alluded to the need —they’re likely to have considerable difficulty addressing ill-defined problems (P. M. King & Kitchener, 2004; Schraw, Dunkle, & Bendixen, 1995). two general categories: algorithms and heuristics. Algorithms As mentioned earlier, well-defined problems can usually be solved using a particular sequence of operations. For example, you can solve the problem 43)3,354 using either of two procedures: (a) apply prescribed methods of long division or (b) push the appropriate sequence of buttons on a calculator or ce. u d o r p ot re n o d , y onl are eproblems s -03 algorithms. u uAlgorithms are typically domain 9 step-by-step proceduresnfor solving called l d 0 a e . 3 u o 2 c s 0 r n 2 . 5 Peuseful with particular problems specific: They’re in3a6 particular content area but are rarely applicable o @ 7 3 3 1 elsewhere. m.nino computer. Either approach leads to the correct answer: 78. Similarly, you can make a tasty pumpkin pie if you follow a recipe to the letter in terms of ingredients, measurements, and oven temperature. Such specific, As an example, young children might use any of several algorithms when given an arithmetic problem such as this one: If I have 2 apples and you give me 4 more apples, how many apples do I have altogether? Children can often solve such problems even if they haven’t yet had formal instruction in addition (T. P. Carpenter & Moser, 1984). An early strategy is simply to put up two fingers and then four additional fingers uce. d o r p e and count all the fingers to reach the solution “6 apples.” Somewhat not rlater, children may begin to use a min o d , y l strategy, in which they start with the larger number problem, they would start with 4) and then on (for 0the9apple 3 e 0 s u u six . . . six apples l dfive, a e . n 3 u o 2 add on, one by one, the smaller number (e.g., counting “four apples . . . then c s 0 r n 2 . e P o365 7 altogether”) (Siegler & Jenkins, 1989). Still1later, of@ course, most children learn the basic addition facts 3 3 no simple addition problems without having to count at all. As new m.tonianswer (e.g., “2 + 4 = 6”) that enable them strategies emerge, children may initially have trouble using them effectively and thus may often resort to the earlier and less efficient, but more dependable, ones. Eventually, they acquire sufficient proficiency with their new strategies that they can comfortably leave the less efficient ones behind (Siegler & Alibali, 2005). Sometimes, when a single algorithm is insufficient to solve a problem, two or more algorithms in combination can lead to a correct solution. But combining algorithms isn’t necessarily as easy as it sounds (Mayer & Wittrock, 1996; Scandura, 1974). In some cases, growing children may have to learn how to combine them e. c u d o r ot rep pro e r t o n do , an y l n o e year-old who knows basic addition and subtraction facts. I (as experimenter) you at a table that has s 03 u.sit u u 9 l d 0 a e n 3 o 2 c 0 toy erspencils, caramel2squares, several kinds of small objectsP (e.g., soldiers, 5.npaper clips, poker chips). I give 6 3 o @ 33I7give you seven pencils—and teach you some you a small number of one kind of object—let’s say 1that o n i n . m “trading rules” for swapping one set of objects for another set. For example, in child-friendly instruction, I through either formal classroom instruction or informal experiences. For example, imagine that you’re a 7- might teach you that if I give you a certain number of caramels, you must give me the same number of toy soldiers plus one more, and if I give you a certain number of toy soldiers, you must give me the same number of pencils plus two more. Algebraically speaking, I’m teaching you these two rules: n caramels = n + 1 toy soldiers e. c u d o r t rep o n o d , I then nlynature, After you’ve mastered several rules ofothis a trade that involves combining 3 ask you.etodmake e 0 s u u 9 l 0 a n 3 u me if I give you a certain number of o two of the rules—for how many should ncgive 202pencilso3 .you 5 Persinstance, 6 @ in this situation, you’re just a 7-year-old. Chances are pretty caramels? Remember now, you’re1 not an7adult 3 3 o n i n m. training in combining algorithms in such a manner (Scandura, 1974). good that you’d need explicit n toy soldiers = n + 2 pencils Heuristics Not all problems can be solved with algorithms. For example, no surefire algorithms exist for getting rid of a junk food addiction or establishing world peace. And in other situations, algorithms may be too time consuming to be practical. As an illustration, here’s an effective but impractical algorithm for determining the best move in a game of checkers: Consider every possible move, then consider every possible next move that the opponent could make in response to each of those moves, then consider every follow-up move that ce. u d o r p e t rthe ountil n o could be made in response to each of those moves, and so on, winner is projected for every d , ly -03 n o e s u an experienced computer 9 conceivable series of moves (Samuel, would take .edeither u onal u1963).2Such 23an-0algorithm c s 0 r n . e 5 P 6 programmer or a lifetime of dedication to a single game oof3checkers. @ 7 3 3 1 .nino When algorithms for a particularm problem are either nonexistent or impractical, people tend to use heuristics —general problem-solving strategies that may or may not yield a correct solution. Following are examples. Talking to oneself about the problem In Chapter 8 , you learned about the value of self-explanation for enhancing long-term memory storage. Self-explanation is often helpful in problem solving as well. Talking to oneself about a problem and the steps ce. u d o r p ot re n o d , y Renkl, 2011; see S. M. Lane & Schooler, 2004, foreano exception). nl 9-03 s u l a -0 5.ncu.edu n 3 o 2 s 0 r 2 e P 36 o Brainstorming @ 7 3 no13 i n . m In brainstorming, one initially tries to generate a large number of possible approaches to a problem without being taken to solve it often enhances one’s ability to identify valid approaches and monitor progress to a solution, provided that the self-explanations are appropriate ones (M. Carr, 2010; Crowley & Siegler, 1999; regard for how realistic or practical they might be. Only after many possibilities have been generated— perhaps including some seemingly bizarre, outlandish ones—are they evaluated for potential usefulness and effectiveness. This postponement of evaluation increases the odds that people will conduct a broad search of long-term memory and possibly stumble on an unusual or creative solution (Baer & Garrett, 2010; Runco & Chand, 1995; Sweller, 2009). uce. d o r p e ot r n o d , y nl into In means–ends analysis, one breaks a problem two or more subproblems and then works successively o 3 e 0 s u 9 l du & Levine, 1982). For example, 0 a e . n 3 u o 2 on each of theme (De Corte et al., 2010; Newell & Simon, 1972; Sweller c s 0 r n 2 . 5 P 36her o @ 7 imagine that an infant sees an attractive toy beyond reach. A string is attached to the toy; its other end is 3 13 o n i n . m at hand. Between the cloth and the infant is a foam rubber barrier. Many 12attached to a cloth closer Means–ends analysis month-old infants can put two and two together and realize that to accomplish the goal (getting the toy), they must first do several other things. Accordingly, they remove the barrier, pull the cloth toward them, grab the string, and reel in the toy (Willatts, 1990; also see Z. Chen, Sanchez, & Campbell, 1997). Working backward e. c u d o r t rep o n o d only, 09-03 e s u l a with certain a formula or geometric configuration characteristics—and asked edtouprove, through a series of . n 3 u o 2 c s 0 r n 2 . e 5 P o36characteristic (the goal) must also be true. mathematically logical steps, how another formula7or another @ 3 3 o1 Sometimes it’s easier—and it’s m just.n asin mathematically valid—to move logically from the goal backward to the In some cases it’s helpful to begin at the problem goal and then work in reverse, one step at a time, toward the initial problem state (Chi & Glaser, 1985; Newell, Shaw, & Simon, 1958; Wickelgren, 1974). Working backward is often applicable to solving algebra and geometry proofs. Students are given an initial situation— initial state. Using visual imagery You’ve previously learned that working memory may include a visuospatial sketchpad that allows short-term storage and manipulation of visual material (see Chapter 7 ). You’ve also learned that visual imagery provides a potentially powerful means of storing information in long-term memory (see Chapter 8 and e. duc o r p e r not o d , y l n e o 3-09-03 s 1999; Kosslyn, 1985; Ormrod, 1979). u u l a n 202 o365.ncu.ed Perso Drawing an analogy 1337@ o n i n . m Chapter 9 ). So when problems are easily visualizable or have an obvious spatial structure, people sometimes use visual imagery to solve them (L. D. English, 1997; Geary, 2006; Hegarty & Kozhevnikov, See if you can solve the following problem: Suppose you are a doctor faced with a patient who has a malignant tumor in his stomach. It is impossible to operate on the patient, but unless the tumor is destroyed the patient will die. There is a kind of ray that can be used to destroy the tumor. If the rays reach the tumor all at once at a sufficiently high intensity, the tumor will be destroyed. Unfortunately, at this intensity the healthy tissue that the rays pass through on the uce. d o r p e not r might be used to destroy the tumor with the they will not affect the tumor either. What y type ofoprocedure d , l on the e s -03 tissue? u 9 l du & Holyoak, 1980, pp. 307–308) rays, and at the same time avoid destroying healthy (Gick 0 a e . n 3 u o 2 c s 0 r n 2 . e P 365 o @ 7 3 13 If you’re having trouble solvingin m.n theoproblem, consider this situation: way to the tumor will also be destroyed. At lower intensities the rays are harmless to healthy tissue, but A general wishes to capture a fortress located in the center of a country. There are many roads radiating outward from the fortress. All have been mined so that, while small groups of men can pass over the roads safely, any large force will detonate the mines. A full-scale direct attack is therefore impossible. The general’s solution is to divide his army into small groups, send each group to the head of a different road, and have the groups converge simultaneously on the fortress. (Gick & Holyoak, 1980, p. 309) ce. u d o r p t re olow-intensity n o idea about how to destroy the tumor: You can shootya,number of rays from different directions, d l n o 3 e -0 students such that they all converge on the tumor are umore likely to solve the tumor 9College l ussimultaneously. d 0 a e . n 3 u o 2 c s 0 r n 2 . e read the solution to the fortress 65 because the two problems can be problem when they’veP first 3problem, o @ 7 3 13 solved in analogous ways (Gick & .Holyoak, m nino 1980). Now go back to the tumor problem. Maybe the general’s strategy in capturing the fortress has given you an Drawing an analogy between a current problem and one or more other, previously solved problems can sometimes provide insights into how the current problem might be addressed (L. D. English, 1997; Leung et al., 2012; Schultz & Lochhead, 1991). For example, a student might solve a math problem by studying a similar, worked-out problem—one that’s been correctly solved using a particular procedure (Mwangi & Sweller, 1998; Reimann & Schult, 1996; Renkl & Atkinson, 2010). As another example, consider a problem that the Greek scientist Archimedes confronted sometime around 250 B.C.: uce. d o r p e tr nocraftsman King Hiero asked a goldsmith to make him a gold crown and gave the the gold he should use. o d , y l n o -03hadccheated When he received the finished crown, hea suspected the-goldsmith use that 9 l duhim by replacing 0 e . n 3 u o 2 s 0 r n 2 . e 365 the goldsmith’s honesty was to some of the gold with silver, P a cheaper metal. The only way too determine @ 7 3 3 1 metal has a particular volume for any given weight, compare the crown’s weight against its .ninoAny mvolume. and that ratio is different for each metal. The crown could be weighed easily enough. But how could its volume be measured? Archimedes was pondering the problem one day as he stepped into a bathtub. He watched the bathwater rise and, through analogy (and perhaps visual imagery as well), immediately identified a solution to the king’s problem: The crown’s volume could be determined by placing it in a container of water and measuring the amount of water that was displaced. e. duc o r p e r not o d , y l n e o 3-09-03 s u u l a n 202 o365.ncu.ed Perso 1337@ o n i n . m . d duce o r p e r not rep t o n o nly, d -03 o e s u al -09 5.ncu.edu n 3 o 2 s r 0 e 2 P 36 o @ 7 3 13 m.nino ce. u d o r p ot re n o d , only 09-03 e s u l a son problems.2023- 365.ncu.edu Analogies are sometimes helpful Pein rsolving 7@o 3 3 1 o n m.nisolution, however: Problem solvers may make an incorrect Using an analogy doesn’t guarantee a correct analogy or draw inappropriate parallels (Bassok, 2003; Mayer & Wittrock, 1996; Novick, 1988). A more significant stumbling block is that, without an expert’s guidance, the chances of retrieving and recognizing a helpful analogy are typically quite slim. People of all ages rarely use analogies to tackle a problem unless its analogue has similar superficial features that make its relevance obvious (Bassok, 2003; Gick & Holyoak, 1980; Holyoak & Koh, 1987; Mayer & Wittrock, 1996). uce. d o r p e not rlimits how much people can do in their heads o d The relatively small capacity of working n memory inevitably , y l o 3 capacity e s -0that u 9 l dustoring some aspects of a problem 0 a while trying to solves ao problem. People can augment by e . n 3 u 2 c 0 r n 2 . e P o365creating a diagram, listing the problem’s externally on either paper or a computer—for example, @ 7 3 3 ino1 viable solutions. An external representation of a problem can also components, or jotting m down .npotentially Using external representations of problem components help people encode the problem more concretely and see interrelationships among its various elements more clearly (Anzai, 1991; De Corte et al., 2010; Fuson & Willis, 1989; Kirsh, 2009). Because many problems have no right or wrong solutions, there may be no single best strategy for solving them. And in any case, different strategies are appropriate in different situations. But sometimes people use strategies at the wrong times, often because they’ve learned such strategies at a rote, meaningless level, as e. c u d o r t rep o n o d ly, -03 Meaningless versus Meaningful Problem onSolving e s u l a -09 5.ncu.edu n 3 o 2 s 0 r 2 e P problem before you read further: o36 See if you can solve this @ 7 3 3 nino1 . m The number of quarters a man has is seven times the number of dimes he has. The value of the dimes we’ll see now. exceeds the value of the quarters by two dollars and fifty cents. How many has he of each coin? (Paige & Simon, 1966, p. 79) If you found an answer to the problem—any answer that might somehow look “right”—you’ve overlooked an e. c u d o r t rep o n o d be solved. (If you tried using algebra to solve this problem—as the first time I saw it—you may have only, I did e s 03 dimes u 9 l du 0 a e . been puzzled to find that your answer involved negative0quantities of bothn and quarters.) n 3 u o 2 c s r . e 2 5 P 6 3 @o their underlying logic, they may 7 3 3 1 When people learn algorithms in a rote manner, without understanding o m.nin important point: Quarters are worth more than dimes. If there are more quarters than dimes, the value of the dimes can’t possibly be greater than the value of the quarters. The problem makes no sense and thus can’t sometimes apply the algorithms unthinkingly and inappropriately (M. Carr, 2010; De Corte et al., 2010; Walkington, Sherman, & Petrosino, 2012). As a result, they may obtain illogical or physically impossible results. Consider the following examples of such meaningless problem solving: A student is asked to figure out how many chickens and how many pigs a farmer has if the farmer has 21 animals with 60 legs in all. The student adds 21 and 60, reasoning that, because the problem says “how many in all,” addition is the logical operation (Lester, 1985). ce. u d o r p ot re n o d , only 09-03 e s u l 1982). a on 23- 365.ncu.edu s 0 r 2 e P Middle school students are asked to calculate o how many 40-person buses are needed to transport 540 @ 7 3 3 1 o majority give an answer that includes a fraction, without acknowledging people to a baseball .ninThe mgame. A student uses subtraction whenever a word problem contains the word left—even when a problem actually requiring addition includes the phrase “John left the room to get more apples” (Schoenfeld, that in the case of buses, only whole numbers are possible (Silver, Shapiro, & Deutsch, 1993). All too often, when schools teach problem solving, they focus on teaching algorithms for well-defined problems but neglect to help students understand why the algorithms work and how they can be used in real-world situations (M. Carr, 2010; Muis & Foy, 2010; Silver et al., 1993; Walkington et al., 2012). For example, perhaps you can recall learning how to solve a long- division problem on paper, but you probably don’t remember learning why you multiply the divisor by each digit in your answer and write the product in a uce. d o r p e t r taught a “keyword” method for solving particular location below the dividend. Or perhaps, as I was, you owere n o d , y …
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