Cognitive Theory of Inquiry Teaching-Collins - Ammon Wiemers ...

\)¡ÞTABLE 8.1Elements that Function as Cases, Independent Variables,and DeÞendent Variables in Different DisciplinesIndependentVariablesDependentVariableArithmeticProblemsNumbers, operators, variable assignmentsAnswersArthistoryPictures, sculpturesTechniques, relation of partsEffects on viewerLawLegal casesLaws, past rulings, facts of caseCourt decisionsMedicineMedical casesSymptoms, history, course of symptomsDiseasesGeographyPlacese. g., latitude, altitude, currentse.g., climateMoral educationSituations, eventsActions, rules of BehaviorFairnessBotanyParticular plantsShapes, leafand branch structureType ofplant*l;äFr*ïí$åååg,íí,ggiugg-Ëgíågååiggå ågííåËåËåågËgåiågåiåå'ågåelåålg*glg,ååËíå,åí ååg*ígååi åågilåågËåågååååå äåå rå ggg åååä gr Fr åï# ååågåå ååå ååå Í

lÞ(¡o,TABLE 8.2Teachers, Students, Domains, and Topicsin the Analyzed TranscriptsAnderson, R. C.Junior-high girlArithmeticDistributive lawAnderson, R. C.Hypothetical studentGeographyFactors affecting temperatureAnderson, R. C.Hypothetical studentMoral educationMorality of American revolutionariesBeberman, M.Junior-high studentsArithmeticNumbers versus numeralsBeberman, M.Junior-high studentsArithmeticAddition of real numbersCollins, A.AdultsGeographyGrain growingCollins, A.AdultsGeographyPopulation densityMentorHypothetical studentMedicineDiagnosis ofdiseaseMiller, A.AdultsLawFaimess ofsentencesSchank, R.Graduate studentsComputer scienceTypes ofplansSocrates (Plato)Slave boyArithmeticArea ofsquaresStevens & CollinsAdultsGeographyCauses ofrainfallWarman, E.PreschoolersMoral educationWho can play with blocksWarman, E.PreschoolersMoral educationCharacter's morality in PeterPanâ^\ ?x\Þ+^F=L¡jõ¡{;-^ø---

258 COLLINS AND STEVENSTABLE 8.3Goals and Subgoals of TeachersL Teach a general rule or theory (e.g., Beberman, Anderson, Collins).a. Debugincorrenthypotheses(e.g.,Bebermanonnumbersandnumerals,socrates,stevensCollins, Feurzeig et al., Anderson on moral education and geography).b. Teach how to make predictions in novel cases (e.g., Beberman, Anderson in arWarman, Collins, Feurzeig et al.).2. Teach how to derive a general rule or theory (e.g., Schank, Vy'arman).a. Teach what questions to ask (e.g., Schank, Warman).b.. Teach what is the natue of a theory (e.g., Schank, Beberman, Stwens & Collirs).c. Teach how to test a rule or theory (e.g., Anderson in geography, Schank).d. Teach students to verbalize and defend rules or theories (e.g., Warman, Miller, Schank)The most frequent goal is for the student to derive a specific rule or theorythe teacher has in mind. For example, in arithmetic Beberman tried to get stuto derive the rule for addition of real numbers, and Anderson the distributive lawIn geography Anderson tried to get the student to understand how distancethe ocean affected temperature, and Stevens and Collins tried to get studentsbuild a first-order theory of the factors affecting rainfall.ilAlong with trying to teach a particular rule or theory, teachers often try to elicitand debug incorrect rules or theories. The teachers want the student to conincorrect hypotheses during learning, so that they will not fall into the same traIater. This kind of goal is evident in Beberman's dialogue in which he tries tothe difference between numbers and numerals, in Socrates' dialogues in whichtraces the consequences ofhis student's hypothesis down to a contradiction, and iAnderson's dialogues on geography and moral education in which he enstudents into revealing their misconceptions.Another goal that frequently pairs with teaching a given rule or theory ising students how to make novel predictions based on the rule or theory.* Simplknowing the structure of a theory is not enough; one must be able to operate on tstructure to deal with new problems. For example, in mathematics Anderson giharder and harder problems for the student to predict the answer.** InCollins and Stevens start with cases that exemplify first-order factors and graduamove to more difficult cases to predict. Feurzeig et al. are trying to get studentsdiagnose novel cases. This goal emphasizes the ability to use the theory one hIearned.The other top-level goal of inquiry teachers is to teach students how to denew rule or theory. For example, Schank tried to get his students to formulatenew theory of planning, and Warman tried to get her preschoolers to devise arule for allocating blocks. Many of the dialogues had a similar aim.*klitor's note: This goal is definitely on an application level of learning (e.g., Gagné'sskills and Gropper's giving and applying expl"anations).x*Editor's note: This is identical to Gropper's .s¿¿p ing treatments (except that Gropper's tspecify detailed ways in which practice examples are (or can be made) easier or harder.8. A COGNITIVE THEOBY OF INQUIRY TEACHING 259One related ability is knowing what questions to askin order to derive a new ruletheory on your own. For example, Warman teaches her preschoolers to evaluaterule by how fair it is. Schank tries to get students to construct a theory bying taxonomic kinds of questions. Feurzeig et al. emphasize consideringt diagnoses before reaching a conclusion.. A goal that underlies many of the dialogues is to teach students whatform a rule;þr theory should take. In Schank's case, the structure of a theory is a set of primi-elements as in chemistry. In one of Beberman's dialogues he taught studentsiithe form of arithmetic rules, where variables replace numbers in-srder to be seneral.vens' and Collins' (1911) notion of a theory of rainfall was a hierarchicallyized, process theory. The principal method for achieving their goal seems toto construct rules or theories of the idealized type.occasionally in the dialogues the teachers pursue a goal of teaching studentsto evaluate a rule or theory that has been constructed. For example, in teachingwhat affects temperature Anderson tried to get the student to learn how tolrol one factor while testing for another. After his students had specified a set ofTirimitive plan types, Schank tried to get rhem to rest out their theory by applying it1tg,a real-world plan (i.e., becoming president). The strategies teachers use areific to the kind of evaluation methods being taught.Finally, it was a clear goal of both Warman and Schank to get their students toize and defend their rules or theories. For example, it is clear why Warman's,hildren were always intemrpting to give their ideas: She was constantly encourag-and rewarding them for joining in. Similarly, Schank tried to get each studentthe class to either offer his or her ideas, adopt one ofthe other's ideas, criticizeof the other's ideas, and so on. Both Warman and Schank stressed the quesingof authority in their dialogues as a means to push students to formulate theirideas.These are the top-level goals we have been able to identify so far. They arein Table 8.3 earlier. In pursuing these goals, teachers adopt supportings of identifying particular omissions or misconceptions and debugging themvens & Collins, 1977). Thus these top-level goals spawn supporting goals thatve the dialogue more locally. This is discussed more fully in the section onstructure.es for Inquiry Teaching.e have decideduse. Thetol0focus on l0 of the most important strategies that inquirystrategies are listed in Table 8.4 together with the teachers;{Mitor's note: It may be helpful to keep in mind that there is a difference between inquiry andapproaches to instruction. Inquíry approaches start with a question, and the remainder ofinstruction attempts to bring the student to an understanding of the answer to that question, byof either an expository approach or a discovery approach. On the other hand, díscoveryhes do not present a rule (or generality) to the student until after the student discovers it. Theì.btudent may be led to discover something without having first formulated a question related to such

260 COLLINS AND STEVENSTABLE 8.4., D¡fferent Instructional Techniquesand their Practitioners1. Selecting positive and negative exempla¡s (Anderson, Miller, Stevens & Collins)2. Yaryingcases systematically (Anderson, Stevens & Collins)3. Selecting counterexamples (Collins, Anderson)4. Generating hypothetical cases (rWarman, Miller)5. Forming hypotheses (Warman, Schank, Anderson, Beberman)6. Testing hypotheses (Anderson, Schank)7. Considering alternative predictions (Feurzeig et al., Warman)8. Entrapping students (Anderson, Collins, Feurzeig et al.)9. Tracing consequences to a contradiction (Socrates, Anderson)10. Questioning authority (Schank, Warman)who used them. Our plan is to show excerpts of the teachers illustratingthese techniques, and then show how the technique can be extended to twodomains.The domains we use to illustrate the techniques are mathematics,moral education, medicine, and law. These domains cover radically dikinds of education: Mathematics exemplifies a highly precise, proceduralmoral education and law exemplify domains in which loosely structuredsystems are paramount (Abelson, 1979), and geography and medicine exdomains in which open-ended, causal knowledge systems are paramount (ColWarnock, Aiello, & Miller, 1975).Setecting Positive and Negative ExemptarsTeachers often choose positive or negative paradigm cases in order tothe relevant factors. Parødigm cøses are cases in which the relevant factorsconsistent with a particular value of the dependent variable. This strategy wæevident in the geographical dialogues of Stevens and Collins (1977), but it isapparent in Anderson's arithmetic dialogue, and Miller's law dialogues.\ùy'e can illustrate this strategy for geography in terms of selecting paradcases for rainfall. In the beginning of their teaching, Stevens and Collinspositive exemplars such as the Amazon, Oregon, and Ireland where all therefactors had values that lead to heavy rainfall. They also chose negative exemlike southern California. northern Africa. and northern Chile where allrelevant factors have values that lead to little rainfall. Only later would they takediscovery, in which case it would not be an inquiry approach to discovery. AlthoughStevens are clearly interested in a model of inquiry instruction, it appears that some of thewhose instruction was analyzed may not have been using an inquiry approach, and that careexercised by a teacher or instructional designer to use the strategies described in this section inway as to implement an inquiry rather than a noninquiry approach to discovery.*Editor's note: These are similar to Gropper's practice of enoß (Chapter 5, p. 139) andmatched nonøramplø (Chapter 9, p.325).8. A COGNITIVE THEORY OF INQUIRY TEACHING 261s like the eastern United States or China where the factors affecting rainfalla more complicated pattern.*method that Anderson used to select cases to illustrate the distributive law intic was based on the strategy of selecting positiveexemplars. Forexample,firstproblemhepresentedwasT x 5 + 3 x 5 = ?Hewantedthestudenttothat because the 5 entered the equation twice, the problem could be easilyved by adding 7 and 3 and multiplying by 5. There are a number of aspects ofs particular problem (and the subsequent problems he gave) that make it aigm case: (l) because 7 and3 addupto 10, the 5 appears as the only significantin the answer; (2) the 5 appears in the same position in both parts of the equation;the 5 is distinct from the other digits in the equation. All these serve to highlightdigit the student must factor out.**his work on discovery learning, Davis (1966) advocated a similar strategy foring cases. In getting students to discover how to solve quadratic equations bying them, he gave problems of the form: X: - 5X * 6 = 0, where the roots3 and2. or * - l2X + 35 = 0. where the roots are 5 and 7. The fact that bothhad the same sign was essential to getting the students to make the correct; only when there are roots of the same sign is it readily apparent that thecient is the sum of the two roots.iÉThis same attempt to pick paradigm cases is apparent in Miller's law dialogues.considering what should be a mandatory sentence for a crime, he considerscases, in which all the relevant factors (e.g., tough guy, repeat offender, nondents) would lead ajudge to give a heavy sentence, and best cases, in whichrelevant factors (e.g., mother with dependents, first offender) would lead tot senteirce. This exactly parallels the Stevens and Collins strategy in geography.jThere are also two other strategies for picking positive and negative exemplarswe have named near hits and near misses after Vy'inston (197r. Near misses'cases in which all the necessary factors but one hold.*** For example, Florida,;fear miss for rice growing, because rice cor¡ld be grow there except for the poorr'Near misses highlight a particular factor that is necessary. Near hits are theirarts for sufficient factors: cases that would not have a particular value ondependent variable, except for the occurrence of a particular sufficient.***t! For example, it is possible to grow rice in Egypt despite little rainfall,of irrigation from the Nile. Near misses and near hits are importantfor highlighting particular necessary or sufflrcient factors.. *Rlitor's note: This is reminiscent of Groppa's shaping.routinæ whersn more difficult examplesbhly praented after easier examples are præented.i t*Editor's note: These a¡e identical to what Gropper refers to æ a.rø (see Chapter 5, p. 128).***Editor's note: This is ideriticål to Menill's notion of a matched nono(omple (see Chapter 9, p. 325)Markle and Tierna¡m's (1969) notion of a clos-in nonøramplar.;f *trEditor's note: This is simila¡ to Merrill's notion of divergent ørnmplø (see Chaptrr 9, p. 3?5),and Tiemannls (l!X9) notion of diveryenc uemplan, and Gropper's notton of varied pracfiæ øc-(see Chapter 5, p. 137).

262 COLLINS AND STEVENS2. Varying Cases SystematicallyTeachers often choose cases in systematic sequences to emphasize particulartors that they want the student to notice. This is most evident in the dialogue iwhich Anderson got a junior-high-school girl to derive the distributive law inarithmetic. He started out giving her problems to work, like 7 X 5 + 3 x 57 x 12 + 3 x 12, in which the only factor that changed was thewhich shows up in the ¿rnswer (50 or 12,0) as the significant digits. He thenproblems in which he varied the addends systematically-7O x 8 + 30 x 86 x 4 + 4 x 4-butpreservedthefactthatthemultiplierformedthedigits. Then he relaxed that corstraint to examples like ll x 6 + 9 x110 x 4 + 90 x 4;andfinally4 x 3 + 8 x 3,sothatthestudentwouldmulate the distributive law in its most general form. Anderson was systemativarying one factor after another in the problems he gave the student, so thatstudent could see how e¿ch factor in tum affected the answer.*We can illustrate this technique in geography by showing how teacherssystematically choose cases to vary the different factors affecting average temature. First, the teacher might systematically vary latitude while holdingvariables constant (e.g., the Amazon jungle, the Pampas, Antarctica), then valtitude while holding other variables constant (e.g., the Amazon jungle, the ciof Quito, the top of Kilimanjaro), then other factors such as distance fromocean, sea and wind currents, cloud and tree cover, and so on. The separationindividual factors in this way is precisely what Anderson was doing in ariIn moral education it is possible to consider what punishment is appropriatelconsidering cases in which the punishable behavior is systematically varied'different respects. For example, the teacher could systematically vary theofthe intention, the severity ofthe act, and the damage ofthe consequences onea time while holdine each of the other factors constant.rIn Collins and Stevens (1982) we point out four different ways this kindsystematic variation can occur. The cases already cited involve differentiation;dffirentiation a set of nonfocused factors is held constant, while the teacherhow variation of one factor affects the dependent variable . Its inverse, genetíon, occurs when the teacher holds the focused factor and the dependent vconstånt, while varying the nonfocused factors.** The two other strategieslight the range of variability of either the focused factor or the dependent variaIn one strategy the teacher holds the focused factor constant while showingwidely the value of the dependent variable may vary (because of variation infocused factors); in the other strategy the teacher holds the dependent variablestant and shows how widely the value of the foatsed factor may vary.*four strategies allow teachers to stress various interactions between different fand the dependent variable.*Editor's note: Again notice the similarity wittr Gropper's vanous shaping routines, especiallyvaríd practice øømplæ (see Chapter 5).+*Editor's note: These two correspond to Gropper's discrimination and generalization.***Editor's note: These hÀ'o are similar to Groppø's varid practiæ øtamplæ (see Chapter Ð.8. A COGNITIVE THEORY OF INOUIRY TEACHING 2633, Selecting CounterexamplesA third method of choosing cases that teachers use in the dialogues we haveis selecting counterexarnples. We can illustrate two different kinds of'r"munterexamples in the following short dialgue on growing rice from Collins,,:(1977):Where in North America do you think rice might be grown?Louisiana.Why there?Places where there is a lot of water. I think rice requires the ability to selectivelyflood fields.OK. Do you think there's a lot of rice in, say, Washington and Oregon?(Counterexample for an insufficient factor)S: Aha, I don't think so.AC: whv?S: There's a lot of water up there too, but there's two reasons. First the climateisn't conducive, and second, I don't think the land is flat enough. You've got tohave flat land so you can flood a lot of it, unless you terrace it.C: What about Japan? (Counterexample for an unnecessary factor)Yeah, well they have this elaborate technology I suppose for terracing land sothey can flood it selectively even though it's tilted overallfirstcounterexample (for an insufficient factor) was chosen because the stuäèntgave rainfall as a sufficient cause ofrice growing. So a place was chosen thal'had a lot of rainfall, but no rice. When the student mentioned mountains as a reason.why no riçe is grown in Oregon, Japan was chosen as a counterexample (for antunnecessary factor), because it is mountainous butproduces rice.* As can be seeni.in the dialogue, counterexamples like these force the student to pay attention toldifferent factors affecting the dependent variable.' 'One can see this same strategy for choosing a counterexample applied to moraleducation in the following excerpt from Anderson (in Collins, 1977):RA: If you'd been alive during the American Revolution, which side would youhave been on?S: The American side.RA: Why?S: They were fighting for their rights.RA: You admire people who fight for their rights. Is that true?S: Yes.RA: How about the young men who broke into the draft office and burned therecords? Do you admire them? (Counterexample for insufficient factors)S: No, what they did was wrong.":lVhat Anderson did was to pick a counterexample for an insufficient factor. Ht,,.rknew the student does not admire everyone who fights for his or her rights, so then'xEditor'snote:Thisisaformoffeedbackonthepractice(see,forexample,Gagné'sseventhinstru(tional event in Chapter 4)-a form in which the student is led to discover his or her own enor.

264 COLLINS AND STEVENS8. A COGNITIVE THEORY OF INQUIRY TEACHING 265must be other factors involved. This line of questioning forces the student to thinkAM: You believe that there should be mandatory sentences? What do you thinkabout some of the different factors that determine the morality of an action.should be the sentence for armed robberv?we can illustrate the use of counterexamples in mathematics with an10 years.flromranaly,tic g:9.."1.y Suppose a studenr hypothesizes on rhe basis of the grapü],So if a hardened criminal robs a bank of $ 1000, he should get 10 years in prisonfor x2t the term;ñ;;; ïi'tiwith no possibility of parole? (Hypothetical case)equation is the radius of the circle. Then the teacher might ask the student toYes that seems fair.the graph of x2 + y2 = 4. The student will find that this yields aWhatoil;;;;if a poor young woman with children, who needs money to feed her kids,rather than radius 4, and may infer that the radius is the square root "ir"l" holds up a grocery store with an unloaded gun. Should she get l0 years too?of the term onthe right. Learning to construct counterexamples is partiãularly useful in matheirmatics, where many proofs and intuitions rest upon this skill.Two kinds of counterexamples were seen in the first excerpt from geography: aWhat Miller does is entrap the man into a confirmation of a harsh rule of sentencingcounterexample for insufficient factors, and one for unnecessary factors. yjthThere,one hypothetical case. His second case faces the man with the oppositecan also be counterexamples for irrelevant factors and incorrect values offactorseltreme (as did Warman's) in which the man's rule is satisfied (armed robbery),(Collins & Stevens, 1982).'ibut in which other factors override the man's evaluation of the fairness of the rule.Eoth Miller and Warman use hypothetical-case construction to force their re-4. Generating Qenerating Hypothetical Casesto take into account other factors in forming a general rule ofbehavior.In the dialogues r of Eloise Warman on moral education and Arthw MillerWe can illustrate how this technique can be extended to geography with anfaimess of sentencing, these teachers often generate hypothetical cases toçxample. Suppose a student thinks rice is grown in Louisiana because it rains a lottheir students' reasoning. warman's use of the strategy was most apparent inre. The teacher might ask "Suppose it didn't rain a lot in Louisiana; could theyclass discussion about a problem that arose because the boys @) in thå chssI grow rice?" In fact, irrigation could be used to grow rice. In the Collins andalways plalnC with the blocks, thus preventing the girls 1C¡ f.o* playmg(1982) paper, we outline four different kinds of hypothetical cases thethem. Two examples of warman's use of thè strategy ocóu¡ in ,ñ" í'"ñä*æcan construct; these parallel the four kinds of counterexamples.excerpt:B: How about no girls play with anything and boys play with everything.5. Forming HypothesesEW: oK' Let's take a vote. Boys, how about if you don'iplay with any tols here inThe most prevalent strategy that teachers use is to get students to formulateschool? Would you like that? (Hypothetical case)gëneral rules relattng different factors to values of the dependent variable. We canB: Noillustrate these attempts in all three domains by excerpts from Beberman, Anderson,and Warman.G: Yea.EW: OK. David said something. What did you say?B: I would In one dialogue Beberman was trying to getstaystudents tohome.formulate a general ruleEW: He would stayl1örhome. OK. Howadditionabout ifof real numbers. To this end he gave students a procedure to workwe had boys could play with every_thing but blocks? (Hypothetical case)i.through on graph paper to add a set of real numbers, by going right for positiveB: No. Ratsþumbers and going left for negative numbers. After a while students found a shortfordoing this: They would.àdd together thewhatpositive numbers, then thewarmannegativedoes systematicaily is to ilrustrate the unfairness of the .ur.ent o, uJproposedand take the difference. Beberman subsequently tried to get them tosituation by reversing the roles as to who gets the advantage. * Thus in theii,foimulate this shortcut procedure into a few general rulesfirstfor adding real numbers,hypothetical case she reverses the boy's propoied rure by suusituting boysgirls. for*!rwhich can be seen in the following dialogue excerpt:In the second hypothetical case she."u"rs", the current situation in which:igirls do not ger ro play wirh the brocks. she reverses the polariry or ,o*"ìà"rårìnlthe situation to force the students to see what factors wili makothings fairer. iA somewhat different version of this strategy is used by Miller in"tris televisionì;S:show Miller's court. In a show on sentencing, iòr example, he carried on ãi"r"g"JìMB:along the following lines with one man (M): " - ìi*Editor's note: Again, this is a form of feedback by discovery.S,MB:I want to state a rule here which would tell somebody how to add negativenumbers if they didn't know how to do it before. Christine?The absolute value-well-d plus b equals uh-negative-Yes, what do we do when we try to do a problem like that? Christine is on theright track. What do you actually do? Go ahead, Christine.You add the numbers of arithmetic 5 and 7 , and then you-I add the numbers of arithmetic 5 and 7; but how do I get the numbers ofarithmetic when I'm talking with pronumerals like this?

266 coLLINs AND STEVENSWe can illustrate the attempt to get students to formulate rules in geographyan excerpt from Anderson (in Collins, 1977) on the factors affecting atemperature. In an earlier part of the dialogue the student had been forced torealization that there were places in the northem hemisphere that were warmerthe average than places to the south of them. The following excerpt shows Anderson(in Collins, 1971) emphasis on hypothesis formation:S:RA:S:RA:S:Some other factor besides north-south distance must also affect temperature.Yes! Right! What could this factor be?I don't have any idea.Why don't you look at your map of North America. Do you see any differencesbetween Montana and Newfoundland?Montana is in the center of the country. Newfoundland is on the ocean.RA: What do you mean by "in the center of the country"?S: It's a long way from the ocean.RA: Do you suppose that distance from the ocean affects temperature?S: I'm not sure. It wouldjust be a guess.RA: True! The name for such a guess is a hypothesis. Supposing the hypothesiswere correct, what exactly would you predict?S: The further a place is from the ocean, the lower the temperature will be in thewinter.In her dialogue on who could play with blocks Warman never explicitlythe children to formulate a new rule, but she stated the problem and encouthem strongly whenever anyone offered a new rule for allocating the blocks.can be seen in the following two short excerpts; in the first Warman rejectsproposed rule because it is the same as the current rule, and in the second shethe rule as the solution to the problem:G:EW:B:EW:I've got a good idea. Everybody play with blocks.What do you think about that?Rats.Isn't that the rule we have right now? That everyone can play with blocks. Butwhat's the problem?B: I've got one idea.EW: Oh, Greg's got a good idea. (Reward rule formulation.)B: The girls can play with the big blocks only on 2 days.EW: Hey, listen we come to school 4 days a week. If the girls play with the bigtiblocks on 2 days that gives the boys 2 other days to play with blocks. Does thatsound fair? (Restate rule. Ask for rule evaluation.)G: Yea! Yea!There are a variety of strategies for prodding students to formulate habout what factors are involved and how they affect the dependent variabl*Editor's note: This is similar to Gropper's notion of .cuing (see Chapter 5).8. A COGNITIVE THEORY OF INQUIRY TEACHING 267are enumerated in Collins and Stevens (1982) as strategies for identifying difelementsin a n¡le or theorv.¡,,,, 6. Evaluating Hypotheses.Sometimes teachers follow up the hypothesis-formation stage by trying to get'3'tudents to systematically test out their hypotheses. This strategy is seen most clear-,¡.þ in the Anderson and the Schank dialogues. Anderson tries to get the student to,test his hypothesis by comparing temperature in different places in the real world.tries to get his students to test out their notions about what are the basici:èlpmenß in planning by applying their taxonomy to a real-world problem, such asi1i-unning for president. We now show how testing hpypotheses c¿n be applied to theprevious examples of hypothesis formation.We start with the Anderson example, in which the student's hypothesis was thatistance from the ocean affects average temperature. The dialogue continuedfollows:RA: How could you test your hypothesis?S: By comparing temperatures ofplaces different distances from the ocean.RA: Very good. Let's do that. Suppose we take St. Louis, Missouri. Which wouldbe best to compare, Atlanta, Georgia, or Washington, D.C.?S: I'm not sure.RA: Why don't you look at your map? Maybe that will help you decide.S: I would pick Washington.RA: Why?S: Because it's at the same latitude as St. Louis.RA: Why is that important?S: Well, if Atlanta were warmer, I wouldn't know whether it was because it wasnearer the ocean or further south.hat Anderson is doing here is teaching the student how to hold other variableswhen testing out a hypothesis. This is also one of the strategies used byiiteachers in the systematic variation of cases described earlier.After Beberman got the students to formulate several rules for the addition of'numbers, he could have had students test their rules by generating widely,different examples to see if the rules as formulated could handle them. For example,rule the class formulated was "If both a and b are negative, add the absoluteof a and the absolute value of å and give the sum a negative sign." Theresuch rules to handle different cases. To test out the rules he could set studentsgenerate different pairs of numbers to see if the rules produce the same answersthe line-drawing procedure. In this case it is particularly important to make surerules work for special cases, such as when a or å equals zero.In the Warman excerpt in which Greg formulates a rule that boys get to play withblocks on 2 days and girls on 2 days, Warman explicitly asks students tothe rule for fairness. This in fact led later to one amendment,to go first because they have been deprived previously. 'Warman that the girlscould have

268 ooLLINS AND STEVENSgone further in evaluating Greg's rule by asking the students to consider its faifor all the people involved: boys, girls, teachers, particular children, and so on.they had done this, they might have amended the rule further to let the child (children) who was playing with the blocks invite one member of the opposite sexplay, because one of the boys had expressed a desire to play with one of the girlsThey could have tested the rule even farther in this situation by trying it outday on which the boys got the blocks half the time and the girls half the time, towhether the new rule worked.There are different aspects to hypothesis evaluation, such as controlling variaor testing out special cases, that are important for students to learn. These canrbrought out by getting students to systematically evaluate their hypotheses.7. Considering Alternative PredictionsHypothesis formation is concerned with identifying different føctorc andthey relate to values of the dependent variable. Thus Anderson was trying tostudents to consider different factors that affect temperature and to specify arelating to factors to temperature. Sometimes, teachers, particularly in theet al. and warman dialogues, try to get the students to consider different alnative values for the dependent variable.*'we can see the teacher trying to get the student to consider alternative predictin the following dialogue on medical diagnosis (Feurzeig et al., 1964):T: We've considered one possibility (i.e., pulmonary infarction). Do you haveanother diagnosis in mind?No.In that case I'd like to talk about viral pneumonia. The tachycardia, high WBC,elevated respiratory rate, shaking chills, bloody sputum, and severe pleuralpain all lend weight to that diagnosis-righr?S:T:What the teacher is doing here is trying to get the student to consider how theof the known factors fit with different possible values of the dependent variabThis forces the student to weigh different alternatives in making any predictjudgments. This same strategy was applied by Collins in his dialogues onfactors affecting grain growing when he asked sfudents to consider whetheror rice or corn could be grown in the same place.An excerpt from Warman illustrates the same strategy applied to moral edtion. The excerpt is"from a dialogue discussing the morality of the dicharacters in Peter Pan, which the children had just seen::EW:S:EW:S:,l,EW:8. A COGNITIVE THEORY OF INQUIRY TEACHING 269Why are the Indians good?No. It's the Chief, because he catched all ofthe boys.So the Chiefcatches all the boys; so is the Chiefgood?Nope. He's bad.He's bad? Is he always bad? Or is he good sometimes, or what do you think?That's a tough question. Is the Indian Chief always bad, or is he sometimesEW: Are the Indians good in Peter Pan?S: Goodbad?What would you say?'warman tries to get the children to consider different points on the moralitynuum, in relation to where the actions of the Indian chief fall on that continuum.:times dependent variables have a discrete set of values as in medicine, andimes they are continuous variables, as in moral education, but in either casel,is possible to get students to consider different possible values.- This strategy can be illustrated in mathematics with an example from geometry.the teacher wantsþnor"the student to figure out what the regular polygon isùÏth the most number of sides that can cover a plane surface. The student mightave decided that four must be the answer, because you can cover a surface withles and squares but not with pentagons. The teacher might press the studentsider six, eight, and twelve as possible answers. part of a mathematician's skillon being able to systematically generate other plausible solutiors and tothey cannot hold.Encouraging students to consider other values ofthe dependent variable forcesn into the more powerful methods of differential diagnosis or comparativeothesis testing as opposed to the more natural tendency to consider only onemativp at a time. x rhis is particularly important to prevent people from jumptoa cÒnclusion without considerinq the best alternative.i,¡i?" Entrapping StudentsThe teachers we have analzyed often use entrapment strategies to get the studentsrcveal their underlying misconceptions. This is most apparent in the dialogues oflerson, collirs, and Feurzeig et al. we can illustrate the use of entrapment indifferent domains by excerpts from three of the dialogues.Anderson frequently uses a kind of entrapment strategy in which he takes thereÍNons and turns them into a general rule.** One example of this ocinthe excerpt in which he formulated the general rule ,,you admire peopleo fight for their rights," and then suggested a counterexample. This strategy c¿nseen later in the same dialogue, when the student defended the American revolu-They were in the right. They didn't have any voice in the government. Therewas taxation without representation*Editor's note: considering that the real content h ere is how to derive a general rule or theoryin Table 8.3), this-like many of the other strategies here-is just a form of practic¿, in whichstudent is required to practice one ofthe parts ofthe algo-heuristic procedures forderiving arule or theory.rlEditorh note: This supports the contention in the preceding Erlitor's note.itor's note: This kind of enÍapment seems to be mostry feedback on practice, where the student,sis the student's response on the practice.

272 COLLINS AND STEVENS8. A COGNITIVE THEORY OF INQUIRY TEACHING 273abcS: No, damn it. They were in the right. They were fighting for their liberty. Theydidn't have any voice in the government. There was taxation without represen-ADBcd::';j, We can illustrate how this same technique can be extended to geography with ançxample from one of Stevens' and Collins' dialogues on the causes ofrainfall inthe Amazon. When asked from where the moisture evaporated that caused theiheavy rainfall in the Amazon jungle, the student incorrectly answered the Amazon' River. The implications of this could be traced with a series of questions such as:r( I ) Does most of the water in the river evaporate or flow into the ocean? (2) If mostOf the water flows into the ocean, won't the process soon dry up? The student will,çickly be forced by this line of reasoning to see that most evaporation must occut:Íf,.rom the ocean rather than from the river.ii).Tracing consequences in this way forces students to actively debug their own@ories. This may prevent students from making similar mistakes in the future,;.,'ilnd it teaches them to evaluate a theory by testing out its consequences.FlG, 8.2 Diagram referred to by Socrates in the Meno dialogue.Boy: Surely it is!Soc: Is four times the old one, double?Boy: Why no, upon mY word!Soc: How big then?Boy: Four times as big!Soc: Then, my boy, from a double line we get a space four times as big, not double.Bov: That's true.What Socrates did was to follow the chain of consequences until the slaverecognized the contradiction.Anderson applied this same strategy in his dialogue comparing the Viedraft resistors to the American revolutionaries. In the segment shown there isseries of four questions in which he traces out several different consequences ofthgstudent's previous statements until the student finally finds a distinction thafldifferentiates the two cases for him:S:I don't think Viet Nam is such a good thing, but you just can't have individualsdeciding which laws they are going to obey.RA: So, you would say the American revolutionaries should have followed the law.S: Yes, I guess so.RA: If they had obediently followed all the laws we might not have had the AmericanRevolution. Is that right?S: Yes.RA: They should have obeyed the laws even if they believed they were unjust. Is thatright?S: I'm not sure. I suppose I have to say yes.RA: In other words what the American revolutionaries did was wrong. That's truê*isn't it?ii'r: 10. Questioning Authority;rA striking aspect of both the Schank and Warman dialogues is the effort thesemake to get the students not to look to the teacher or the book for the corstrategyto Schank's and Warman's goals of teaching students how toli@t answers, but rather to construct their own theories. This is a particularly im-thei¡ own rules or theories.We can illustrate how Schank and 'Warman apply the strategy with shortxcerpts from their class sessions. The segment from Schank shows him trying toþstudents to form a taxonomy of basic types of plans. He complains when heiiiecognizes that the students are just repeating what they read in the book:RS: Give me some categories of plans.',.,;.t 52: Bargain object. (laughter)I RS: Give me a better one than that. Anyway it's not a category, that's a plan.i, S1: Plans to obtain objects. (Schank writes it down).r., 54: Is there a reason why we want that category?i:"Írll: RS: No, I'mjust looking for gross categories.l;:l52: Plans to establish social control over something.ii:RS: The two of you are agreeing that everything from the book is gospel. It's allright. Give me something new-I wrote those-invent something.Warman comes to the same problem in her dialogue when she is trying to get herers to develop a new rule to decide who is allowed to play with the toyin the classroom. The current rule is that anybody can play with anything,the boys are dominating the use of the blocks, thus keeping the girls from them.following excerpt shows her argument against deciding by authority:EW: Do you think that it should be all right for only one person should get to makeall the choices (sic) for who gets to play wirh blocks. Or do you think it shouldbe somethins we all decide on?

274 ooLLINS AND STEVENSG: I think it should be the teacher's.EW: But why just the teacher's? It doesn't seem to work. We had an idea. We'vebeen trying.The questioning of authority is an important strategy for getting srudents tolike scientists, to get them to try out theory construction on their own, and tothem to question those things that may appear to be givens.Dialogue Control StructureThe control structure that the teacher uses to allocate time between differentand subgoals may be the most crucial aspect for effective teaching.* An earliattempt at a theory of the control structure was developed in stevens and colli(1977), based on tutors' comments about what they thought a student kneweach answer and about why they asked each question. The four basic parts of,control-structure theory are: (l) a set of strategies for selecting cases withto the top-level goals; (2) a student model; (3) an agenda; and (4) a set ofrules f.or adding goals and subgoals to the agenda.Given a set of topJevel goals, the teacher selects cases that,optimize the abiof the student to master those goals. There appear to be several overallthat the teachers apply in selecting cases:1. Selecting cases that illustrate more important factors beþre lessfactors. For example, in teaching about rainfall, Stevens and collins movecases like the Amazon and Ireland that exemplify a first-order theory to cases lieåstem America or Patagonia where the factors are more complex.**2. select cøsæ to movefrom concrete to abstractfactors. Teachers tend toc¿ses that emphasize concrete factors initially, in order to make contact withstudent's experience, and then move to cases that emphasize morefaCtOfS, ¡1. '¡ lc3. select more important or more frequent cases beþre less important orfrequent cases. other things being equal , a geography teacher will select cases Ithe United States, Europe, and China, which are more important. Aprofessor will select the most frequent diseases and the ones that are most ito diagnose.*Fdiûor's note: To a large o$ent the authors are talking here about tle importaqnce of a,søst¡ategy. A,sequencing strategy is also involved to some oderit (see following discussion).r*Blito¡'s note: This is a bit unclea¡ because importance and complotity a¡e two differenr Iwith respect to complexity, this prescription is simila¡ to Gropper's shaping routinø and sanelaboration on the simplnt path. With rcpect to importance, it is similar to the Elaboration Theorvstruction (see Chapter 10), except that the latter is a macro-level strategy.iiiEditor's note: This is similar to cropper''s Øttcrete and abstract stimuhts/twonæ mdsChapter 5, pp. 135-137).8. A COGNITIVE THEORY OF INQUIRY TEACHING 275i,., When a case is selected, the teacher begins questioning students about the valuesof.the dependent and independent variables, and the rules interrelating them. Thewers reveal what the student does and does not know with respect to theher's theory (Stevens & Collins, 1917).x As the teacher gains informationt the student's understanding, factors in the teacher's theory are tagged aswn, in error, not known, and so on. This is the basic student model.i, The teacher's model of the student also includes a priori expectations of howllikely any student is to know a given piece of information in the theory (Collins,, & Passafiume, t975). As a particular student reveals what he or shehis or her level of sophistication with respect to the teacher's theory can beFrom this, an estimate can be made as to the likelihood that the studentknow any given factor in the theory. This enables the teacher to focus oninformation near the edge of what the student knows a p¡ori. a¡s details ofthis operates are given in Collins, Wamock, and Passafiume (1975).As specific bugs (i . e. , errors and omissions) in the student's theory or reasoningare identified, they create subgoals to diagnose the underlying causes ofbug and to correct them. Often the questions reveal multiple bugs. In suchthe teacher can only pursue one bug at a time. Thus there has to be an ø genda,orders the subgoals according to which will be pursued first, second, third,so on,**trn adding subgoals to the agenda, there must be a set of priority rules. The priori-;(ies we found in the earlier work (Stevens & Collins, 1977) were:t'iz.a J..4.Errors before omissions.Prior steps before later steps.***Shorter fixes before longer fixes.More important factors before less important factors.iences. Prior steps take priority because the teacher wants to take things up in atake priority over omissions because they have more devastating conse-lional order, to the degree the order is not determined by the student's responses.fixes, like telling the student the right answer, take priority because theyeasier to complete. More important factors take priority because of the orderplied by the overall goals.When more than one bug has been diagnosed, the teacher holds all but the oneed on the agenda, in order of their priority. When the teacher has fixed one*Editor's note: This serves the same function as Scandura's procedures for assæsing individual(Chapter 7, p. Z0). It has the same limitation of only being applicable in individualized instrucrconlo(ts.**Editor's note: This is clearly a sequencing strategf.*Rlitor's note: This is simila¡ to Groppr's forward chaining. As cropper argues, this may be theimportant criterion.

276 COLLINS AND STEVENSbug, he or she takes up the next highest priority bug, and attempts to fixSometimes when trying to fix one bug, the teacher diagnoses another bug. Ifnew bug is of a higher priority, the teacher sometimes interrupts the goal he oris pursuing to fix the higherpriority bug. Thus in the dialogues, there is apatterndiagnosing bugs at different times and holding them there until there is timecorrect them.CONCLUSIONThese techniques of inquiry teaching are designed to teach students torules and theories by dealing with specific cases,* and to apply these rulestheories to new cases. In this process the student is learning two kinds oft(l) specific theories about the knowledge domain; and (2) a variety ofskills (see Table 8 .3) . In some sense the inquiry method models for the studenfprocess ofbeing a scientist.The kinds of reasoning skills we think the student learns from this processforming hypotheses, testing hypotheses, making predictions, selecting optiÇases to test a theory, generating counterexamples and hypothetical cases, disguishing between necessary and sufficient conditions, considering alhypotheses, knowing the forms that rules and theories can take, knowingquestions to ask, and so on (see Table 8.4). In short, all the reasoning skillsscientists need arise in this model of teaching.Furthermore the technique is exceptionally motivating for students.**become involved in the process of creating new theories or recreating theorieshave been developed over centuries. It can be an exhilarating experience forstudents.In summary, by turning learning into problem solving, by carefully seleccases that optimize the abilities the teacher is trying to teach, by makinggrapple with counterexamples and entrapments, teachers challenge the studmore than by any other teaching method. The students come out of the experirable to attack novel problems by applying these strategies themselves.ACKNOWLEDGMENTSThis resea¡ch was sponsored by the Personnel and Training Researdr Programs,logical Sciences Division. Office of Naval Research under ContractNml4-79-C{338. Cont¡act Authority Identification Number NRI 54428.*Editor's note: This is what Landa refers to as discovery by ørnmplæ (w Ctrapta 6, p. 178),++Editor's note: See Chapter 11 for a description of some ofthe reasons why.8. A COGNITIVE THEORY OF INQUIRY TEACHING 277We thank Dean Russell Zwoyer of the School of Education at the Universitv of Illinois,ign-Urbana, for sending us transcripts of Professor Max Beberman, Dr. IrvingI of the Educational Testing Service for sending us a tape and transcript of Eloise, Dr. Richard C. Anderson for sending us various tapes, transcripts, and otherls illustrating Socratic tutoring, and Natalie Dehn for recording Professor Rogerduring one ofhis classes at Yale.REFERENCES,R. P. Differences between belief and knowledge systems. Cognitive Science, 19'79, 3,385.R. C., & Faust, G. \N . Educational psychology : The science of instruction and learning.'.ÌNêw York: Dodd, Mead, 1974n,J.S.,&Burton,R.R. Diagnosticmodelsforproceduralbugsinbasicmathematicalskillsit:Cognitive S c ie nce, 197 8, 2, | 55 - 192, W. G., & Simon, H. A. The mind's eye in chess. In W. G. Chase (Ed.), Visual informationprocessing. New York: Academic Press, 1973., A. Processes in acquiring knowledge. In R. C. Anderson, R. J. Spiro, & W. E. Montague' @ds.), Schooling and the acquisition of knowledg¿. Hillsdale, N.J.: Lawrence Erlbaum Associates,ins, A. Explicating the tacit knowledge in teaching and learning. Paper presented at AmericantEducational Research Association, Toronto, I 9784., & Stevens, A. L. Goals and srategies of effective teachers. In R. Glaser (EÀ.), Advancesilin instructiorul psychologlt (Vol. 2). Flillsdale, N.J.: l¿wrence Erlbaum Associates, 1982.ins, 4., Wamock, E. H., Aiello, N., & Miller, M. L. Reasoning from incomplete knowledge.In D. Bobrow & A. Collins (F,ds.), Representation and understanding: Studies in cognitive'3c¡enc¿. New York: Academic Press. 1975ins,4., Wamock, E. H., & Passafiume, J. J. Analysis and synthesis of tutorial dialogues. In'lG. H. Bower (Ed.), The psychology of learning and motivation (Vol. 9). New York: Academic,R.B. Discoveryintheteachingofmathematics. InL.S.Shulman&E.R.Keisler(Eds.),'''Learning by discovery: A critical appraisal. Chicago: Rand McNally, 1966ig, W., Munter, P., Swets, J., & Breen, M. Computer-aided teaching in medical diagnosisJ.ournal of Medical Education. 1964. 3 9.'7 46-'7 55.in, J. Information processing models and science instruction. In J. Lochhead & J. Clement (Eds.),.Cognitiveprocessinstruction. Philadelphia:TheFranklinlnstitutePress,1979, A. Sentencing. Shown on Miller's Cour\, WCVB, Boston on October 12, 1979.4., & Simon, H. A.. Human problerh solving. EnSewood Cliffs, N.J.: Prentice-Hall,Lachés, Protagoras, Meno, and Euthydemusaward University Press, 1924.; I. E., & Saunders, R. An inquiry into inquiry: Question-asking as an instructional model. InKatz (Ed. ), Current topics in early childhood education. Norwood, N. J. : Ablex, 1979.;D.P.,&Simon,H.A.,Ataleoftwoprotocols. InJ.Lochhead&J.Clement(Eds.),Cognitiveinstruction. Philadelphia: The Franklin Institute Press, 1979.,4.L.,&Collins,A. Thegoal structureofaSocraticfutor.ProceedingsofAssociationforQòmputing M achinery National C onference,Seattle, Washington, |9'17A. L., & Collins, A. Multiple conceptual models of a complex system. In R. Snow,Federico, & W. Montague (Eds.), Aptitude, learning, and instruction: Cognitive process. Hillsdale, N.J.: Lawrence Erlbaum Associates, 1980.

278COLLINS AND STEVENSStevens, A. L., Collins, 4., & Goldin, S. Misconceptions in studend understanding.Joumal of Man-Machine Studi*, ln9, II, 145-156.Vanlehn,K.,&Brown,J.S. Planningnets:Arepresentationforformalizinganalogiesandsemanmodels of procedural skills. In R. Snow, P. Federico, & W. Montague (Eds.), Aptitude,and instruction: Cognitive process analysis. Hillsdale, N.J.: Lawrence Erlbaum Associates, IWinston, P. Leaming to identify toy block structures. In R. L. Solso (Ed.), Contemporary issuèscognitive psychology: The Loyola Symposium. Washington, D.C.: Winston, 1973.Component Display Theory; David Merrillof Southern Californiaú,,'tiþ'{...M. David Merrill,M. David Merrill, Professor of Educational Psychology and Technology at[e University of Southern California, is an internationally recognized inctionalresearch scientist. His work includes major theoretical posioninstructional strategies (as represented by Component Displaydescribed in the chapter) and Content Structure (as represented byboration Theory as described in the chapter by Charles M. Reigeluth). Heauthored or coauthored 9 books related to instructional design and hashored or coauthored 10 chapters in edited books. He and his associatesco.mpleted more than 100 research studies on instructional processes,of which have appeared in educational research journals.. Merrill received the Ph.D. degree in 1964 from the University of lllinois.has served on the faculty of George Peabody College for TeachersBrigham Young University (1966-67; 19ô8-79), Stanford Universi-t'(1967-68) and the University of Southern California (1979-present). AtDr. Merrill founded and directed the Division of Instructional Research,