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Constructing ConstructivistTeaching: Reflection as researchBonita C. WhitePublished online: 18 Aug 2010.
To cite this article: Bonita C. White (2002) Constructing Constructivist Teaching:Reflection as research, Reflective Practice: International and MultidisciplinaryPerspectives, 3:3, 307-326, DOI: 10.1080/1462394022000034550
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Re� ective Practice, Vol. 3, No. 3, 2002
Constructing Constructivist Teaching:re� ection as researchBONITA C. WHITEDepartment of Instruction and Curriculum Leadership, Education 404, University ofMemphis, Memphis, TN 38152, USA
ABSTRACT The author set out to learn how to be a constructivist teacher in an elementarymathematics methods course, by recording in questions the aspects of teaching with whichshe was grappling. She then analyzed the questions in terms of content and process. Thequestion content centered on task creation, content of her goals for teaching, assessment andways to parse knowledge for assessment purposes, and cognition. The process contained thefollowing patterns: (a) questions were revisited with particular foci repeatedly; (b) thequestions moved through technical, interpretative, and critical rationality; (c) concernabout the epistemology and beliefs pre-service teachers hold about teaching undergirded mostof the interpretative and critical rationality questions; and (d) the questions became criticalrational only after the author had acquired considerable knowledge of the complexityinherent in her teaching situation and only after she became aware of issues imbedded in herquestions, the solutions of which would likely affect some students positively but would affectothers negatively.
It was easy to talk about constructivist teaching as long as it was just rhetoric. Thewords rolled past one another effortlessly. Completely unaware of the super� cialityof understanding, I could recite the merits of active learning, of students construct-ing their own knowledge, of teachers as facilitators, and on and on ad nauseam. Littledid I know—literally.
Graduating from university with a brand new PhD and having only recently takencourses in mathematics education, all of which extolled the Curriculum and Evalu-ation Standards (1989), I had become a convert to constructivism, a way of thinkingabout teaching and learning new for me. I had 25 years’ teaching experience, spreadabout evenly between middle and secondary levels. Major teaching areas had beenboth mathematics and language arts. I was well versed in the practice of teaching.Furthermore, I was familiar with the prominent theories of learning from course-work completed in my studies. In short, I was able to recognize the commendablepedagogy intrinsic to the Standards. I was anxious to put constructivism into effectin my own instruction and was given the opportunity when hired to be theelementary mathematics professor at a midsize private university in the Midwest. At
ISSN 1462-3943 print; 1470-1103 online/02/030307-20 Ó 2002 Taylor & Francis LtdDOI: 10.1080/1462394022000034550
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the beginning of my � rst semester of teaching, I began making a conscious attemptto ‘restructure’ my teaching, moving away from transmissive and toward construc-tivist.
Teaching new content at a higher level to a different population of students in anunfamiliar educational culture, using transitional pedagogy, I was in a state of fairlyacute discomfort for several semesters. Aware of the characteristics of noviceteachers, I was able to identify those same traits in myself. I was unable to discernwhich parts of the content were more and less important, and had only surfaceunderstanding of the characteristics of post-secondary students. I gave up the onereal strength I brought to the position—an ability to ‘tell’ in ways that rendered thecontent understandable by students. I did so because I believed pedagogy that wasconstructivist would bene� t students in ways other than acquisition of facts andskills. Being of adventurous spirit, having con� dence in my ability to learn, andbeing intolerant of a lack of teaching expertise in myself, I set my sights on learninghow to teach so that my students could construct understandings rather thanmemorize facts.
My operating assumption was that constructivism translated to inquiry shared bythe teacher and the students. I interpreted this to mean I would teach throughquestioning and my students would be active learners who would have a sense ofownership of that which they constructed. Such ideals, though lofty, were too vagueto inform actual practice. Finding myself in the middle of a class peopled bystudents and content, I was uncertain what speci� c actions to take that might beconstructivist in nature, or when to take them. Knowing what not to do did little tonothing to inform me about what to do. My teaching was analogous to trying to walkon quicksand. I had no lodestone from which I could launch my teaching to beginto establish a foundation from which to operate. Most of the students in myelementary mathematics methods class became frustrated with me, saying I wasunclear and did not provide adequate leadership or direction. Frustration for themtranslated into anxiety for me. At times my resolve weakened, and I was tempted toreturn to transmissive teaching—a zone of comfort for both my students and me. Mybelief in the power of student ownership of ideas was strong, however, so I persisted.
Deconstructing Re� ection (Data Analysis) and Deconstructed Re� ection(Findings)
I had begun my exploration by posing questions for myself, the � rst of which was,‘When should I answer questions, and when should I re� ect them back to the asker?’By the time I had posed for myself a third question, I realized that I was building myunderstanding of what it means to construct constructivist teaching one issue at atime, and that it might be of interest to record traces of that journey en route. Irecorded the third question, preceded by the � rst two (which I remembered clearlybecause I was still in the process of answering them). Thereafter, I documented theissues with which I was grappling, along with a very succinct description (2–4sentences) of my thinking about them, as soon as I became aware that I was working
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Constructing Constructivist Teaching 309
on a new issue, though previous ones were often unresolved. True to my conceptionof constructivist thinking, I thought of and recorded the issues as questions.
Four years later, I had recorded 18 questions that varied in length from a sentenceor two to nearly a page. (See Appendix A for a complete listing of the questions intheir original form.) Aware that the list was un� nished—indeed, was unlikely ever tobe � nished—I decided to take a look at my questions to see what, if anything, Icould learn from them. During analysis, I looked at the content of the questions andat the process through which they progressed.
Content of the Question
My initial reaction to my questions was surprise at what I had written. Because Ideliberately had not reviewed the questions each time I opened the � le to add a newquestion (other than the � rst 2–3, which appeared on the screen each time I openedthe � le), and because my mind was always immersed in the question(s) on which Iwas working at the time, I had no conscious awareness of their content or of the � owthey had taken. I made no changes in them (other than basic grammatical editing)in either content or the order in which they were written before commencinganalysis.
I became aware immediately that the questions subsequent to the � rst 4–6 becameincreasingly elaborated. Performing basic content analysis of the questions, I foundthat each centered on some single issue. My unit of analysis (Lincoln & Guba, 1985)became the question most centrally related to the issue. Again surprised, I discov-ered that the concepts named during analysis appeared repeatedly, making possibletheir categorization. The questions fell into four categories. The categories and thenumber of the questions in each are: task (1, 3, 4, 9, 15), content (2, 6, 12, 13, 14,16, 18), assessment/knowledge (7, 8, 11, 17), and cognition (5, 10). Task consistedof those things in which I asked students to engage, whether to answer a questionin class, perform an out-of-class assignment, or participate in an in-class instruc-tional activity. Content consisted of goals for instruction, including elementarymathematics, pedagogy, and student attitude/disposition, or teaching (the actualiza-tion of the � rst three). Assessment and knowledge were typically paired in questions.Assessment focused on what I can know about what students know and includedgrading issues. Knowledge was content taken as a whole, i.e. mathematics, peda-gogy, disposition, or teaching, and my focus was consistently on ways to parse it intosome typology. Cognition had to do with how learning occurs. I have included threeexamples at levels of increasing complexity. They are embedded in the � ndings forlevels of rationality below.
Depth of Understanding of the Problem
Each time I revisited a question topic (for example, task), I reframed it, establishingdifferent constraints for it than I had done previously. To get a sense of thetransformation in my thinking that was occurring as I reframed, I utilized under-standings I had gained from Dewey (1933). Dewey described re� ective thinking as
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problematizing generally accepted mores of knowledge or practice, and as active,persistent, and careful consideration’ (p. 9) of the situations problematized (Dewey,1933). Innate tendencies he spoke of as making re� ection possible and of affectingits quality are curiosity, suggestion, and orderliness of thinking. The ideas onsuggestion informed my analysis of my level of understanding of the situation asrepresented in my questions.
Suggestions, according to Dewey (1933) are ideas that come to mind unbidden.They have their source in prior experiences or thoughts. Affecting the quality ofsuggestion are ease, variety, and depth. These affect all aspects of re� ection,including problem posing. I applied Dewey’s (1933) conception of depth of sugges-tion to better understand the complexity of my questions or problem posing.
Dewey (1933) wrote of depth or profundity of thought, ‘We distinguish betweenpeople not only upon the basis of the quickness and variety of their intellectualresponse, but also with respect to the plane upon which these occur—the intrinsicquality of their response’ (p. 44). I judged the depth of my understanding of thecontext of the question/problem in terms of the speci� city of the de� ning boundariesI placed on them. To distinguish them, I have labeled the two types of de� ningboundaries ‘conditions’ and ‘contingencies’. Conditions are the speci� cations thatserve to narrow the focus of the question; they are the ‘givens’ of the question/prob-lem. Contingencies are circumstances for which more than one possible enactmentexists, each of which would affect the outcome differently. My assumption is thatdepth of understanding is re� ected in the speci� city of the conditions placed on theproblem statement and in the number of contingencies considered. To illustrate, Ihave unitized (Lincoln & Guba, 1985) Question 14 into the analyzed portions of thequestion below. I have altered the question in no way except to insert the terms‘question’, ‘condition’ and ‘contingency’ in front of the statement or series of statementsto which the term applies.
Question: What balance do I strike between helping students learn contentand pedagogy and helping students acquire (or further develop) the dispo-sition to seek out and re� ectively consider the problems of teaching?Condition: This disposition stands in opposition to one in which studentsdo not see problems that exist or who view problems as simple and ashaving prescriptive answers. Contingency: The problem here is that disposi-tional change requires that considerable class time be devoted to theanalysis of problematic situations rather than learning the knowledgebase—whether that consists of teaching strategies or knowledge of con-tent—but we already lack time enough in one semester to teach the contentor pedagogy. Condition: Confounding that issue is the fact that the studentsmust possess a knowledge base of content and pedagogy to be functionalin student teaching.
The central question is the � rst statement, i.e. what balance should I try to attainbetween helping students learn content and pedagody and helping them developdispositions. The conditions of the question are that the students must have contentand pedagogical knowledge, and that students often do not see problems that exist
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Constructing Constructivist Teaching 311
or view them super� cially. The contingency that will affect the outcome of thequestion is time, i.e. considerable time is necessary to help students develop thedisposition to view problems as complex.
The progression through which my questions move reveals an increase in theknowledge I use to inform the question and an increase in my understanding of thecomplexity of the context in which the problem is situated. I have embeddedexamples of questions at increasing levels of depth of understanding in the � ndingsfor levels of rationality below.
Process through which the Questions/Problems Progress
Having become aware that my understanding of the context of my teaching hasgrown increasingly deeper, I sought some framework that would allow me to codifymy understanding. I needed some label that would provide a consistent vehicle forunderstanding the learning process through which I was moving. RememberingHabermas (1974) as conceptually related to Dewey (1933) and remembering VanManen’s (1977) translation of his ideas of critical social theory to education, Iconsidered those ideas.
Whereas Dewey described aspects of the thinking process involved with re� ection,Van Manen (1977) described it in terms of what becomes problematized. ApplyingHabermas’s (1974) critical social theory to education, Van Manen (1977) madeHabermas’s work broadly accessible to educators. He (Van Manen, 1977) describesrationality in terms of three levels: technical, interpretative, and critical rational.
At level one, the Technical Rational level, means–ends thinking is dominant.What is problematic, however, is the means; ends are accepted as givens, i.e. whatis to be accomplished is not in question, only how to accomplish it. Put another way,the end necessarily results from some eliciting cause that is part of the natural orderof things, but the means can be affected for better or for worse. Underlying this viewis that people are invariant—individual differences do not affect outcome.
Thinking at the Technical Rational level functions to produce sets of principles forhow to do that give rise to corresponding theories and recommendations forapplication. Because the means—not the end—are in question and because peopleare considered invariant, the values that guide enactment and that serve as criteriafor judging quality are ef� ciency and effectiveness. Technical Rational thinkingbecomes challenged when the thinker realizes that sometimes multiple principles canbe applied to a single situation, and the outcome for each will differ from theoutcome of the others, though all may be exemplars of ef� ciency and effectiveness.The thinker becomes aware of the need for some way of deciding between compet-ing principles, some overarching theory that will provide criteria to guide theselection of principle.
The need to � nd a way to choose among competing principles propels the thinkerinto level two rationality—Interpretative Rationality. Attempting to sort throughdifferent ends and the principles through which they were elicited, the re� ectivethinker must turn attention to the inner workings of the principles-in-action, i.e. theproblematic becomes the internal workings of each principle-in-action. Inexorably,
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the focus of attention is drawn to the reactions of the people involved. Hence, theproblematic becomes the making visible and understandable (in an existential sense)of the educational experiences of all participants. Coming to understand in thiscontext requires open communication, and it requires that the re� ective thinker beable to immerse the self in separate realities, i.e. be able to orient his or her thinkingto view phenomena from some (participant’s) perspective and to think the ways ofthinking and act the ways of being operant within that perspective, and then to dothe same mental feats within another perspective, and another and another, all ofwhich may be different from their own preferred way of thinking and being. Thus,the re� ective thinker gains awareness of the values held by different participants andby different orientations (groups of participants).
Understanding that value systems differ and that different means result in differ-ent ends, including the goods of the society, makes possible comparison acrossparticipants and orientations, which serves to call into question things previouslyunquestioned. An end that is valued within one perspective or orientation may notbe valued within another or may even be immoral from the point of view of a third.Likewise, the goods of a society may be accessible to some but not to others and mayeven be gained at the exploitation of others. Thus do the ends assumed at theTechnical Rational level and the means for attaining come to be placed in oppositionto one another.
The realization that different means result in different ends for various partici-pants (and orientations) serves as the springboard into level three rationality, CriticalRationality, in which all aspects of reality become problematic. Understanding thatin Critical Rationality all aspects of reality are grist for scrutiny, Van Manen (1977)writes, ‘unlimited inquiry, a constant critique, and a fundamental self-criticism isvital’ (p. 221). Having attained an understanding of the relative nature of reality,critically rational re� ective thinkers are also aware of their reciprocal relationshipwith the socioculture of which they are a part, ‘as knowing subjects rather than asrecipients, [Critical Rational thinkers] achieve a deepening awareness both of thesociocultural reality that shapes their lives and of their capacity to transform thatreality’ (Van Manen, 1977, p. 222).
At this level of thinking, the re� ective person is impelled to search for some set ofvalues that will guide decision-making. Though this question is an open one,Habermas (1974) recommends values that have long been a theme of the Westernphilosophical tradition—the creation of a society that is just and fair. He explainswhat that means: ‘Universal consensus, free from delusions or distortions, is theideal of a deliberative rationality that pursues worthwhile educational ends inself-determination, community, and on the basis of justice, equality, and freedom’(Habermas, 1974, p. 227). This ideal has been broadly accepted by researchers ineducation.
My questions appeared suitable for interpretation through Van Manen’s (1977)interpretation of rationality because they were formatted as questions and theyrevealed increasing depth of understanding. My next step in data analysis was toconsider my questions in terms of level of rationality. I found questions at all threelevels.
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Constructing Constructivist Teaching 313
Below I describe an example question/problem at each level in terms of itscomplexity and in terms of its rationality. My description includes the conditionsand the contingencies I place on an answer to the question. Conditions aredelimiting speci� cations of circumstances that might affect an answer. I haveincluded the number of the sample question. In front of each statement is anitalicized identi� er (question, condition, contingency). These descriptors separate thequestion into the various identi� ed elements. I have also labeled additional andrelated questions of which I gained awareness in the process of articulating thecentral question. Except for the addition of italicized identi� ers, the questions areunchanged from their original form. Following these examples of the three levels ofrationality, I describe two salient patterns that emerged in question sequence: (a)beliefs as a motivating concern, and (b) catalyst for Critical Rational thinking.
Level 1: Technical rationality. Little knowledge informed my initial questions. SeeNumbers 1–6 in Appendix A. They had only the broadest conditions delimitingtheir boundaries. At this point, my questions were what Van Manen (1977) calledTechnical Rational in nature, i.e. they focused on what to do and how and when todo it for the purpose of attaining a given end. In the example below, my reason forwanting to attain the end is a given in the task I set for myself, i.e. the desired endis constructivist pedagogy. I articulate neither conditions nor contingencies in thisquestion. The closest I am able to come to identifying a condition is what should notbe the case.
Question: What should get constructed? Condition: I know it doesn’t makesense to have students try to construct something procedural like how touse a calculator, but Question repeated: I don’t know what they shouldconstruct. (Q. 2)
This question reveals that I know what procedural knowledge is, and that I am ableto cite at least one example of mathematical procedural knowledge. My inability toarticulate conditions or contingencies for the question reveals that my understandingof the problematic situation is highly super� cial at this point (Dewey, 1933).
Level 2: Transition from Technical to Interpretative Rationality. In Question 7, Itransition from Technical Rational to Interpretative Rationality. Having attainedsome equanimity (during the process of resolving questions 1–6) in my understand-ing of the ways students might come to construct understandings of the facts,concepts, and procedures of elementary mathematics, I turned my attention toassessment. This transition occurs as a result of cognitive dissonance arising out ofmy understandings of the theoretical description of construction of knowledge andmy understanding of assessment, particularly alignment of the assessment task withthe construct it reputes to assess.
That this question is Interpretative Rationality is a result of my having immersedmyself in the separate realities of constructivism and assessment. I am able to thinkin both modes or from both perspectives. Doing so in the practical situation of
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teaching calls into question how the tenets of both can be in con� ict with oneanother and still be ‘best.’ Following is Question 7:
Question: How do I assess what students construct? Condition 1: If con-struction, by de� nition, requires that what’s learned be connected to what’sknown, and Condition 2: if what’s known is different for each person,Revised and more focused Question: how can I devise assessment tasks thatwill let me know if the students really do know what I am trying to assess?Contingency 1: What if the students connect the knowledge to somethingpersonal as opposed to other mathematical knowledge? Contingency 2:What if they interweave what they construct with what’s already known sothat something new and different has been created? Question repeated: Howcan I assess their grasp of the knowledge I want to assess? New question:Would it ever be possible for me to assign different letter grades to studentswho have (the same) knowledge that is constructed in personal ways, and,Condition 1 for new question: if I cannot do so, New Question: what is thepurpose of formal assessment? (Q. 7)
This question is far better elaborated than earlier ones. It contains two conditionsand two contingencies. The conditions reveal a newly acquired question about therelationship between formal assessment and constructed knowledge that leads to arelated question about grading. My understanding of the problematic situation hasgrown de� nitely deeper than it was at the beginning.
Level 3: Transition from Interpretative Rationality to Critical Rationality. In Question13, I transition into Critical Rationality. I have become aware that differencessometimes exist between separate perspectives that place them in con� ict with oneanother. The fact that each perspective poses its own demands on practice (VanManen, 1977) has fomented in me an awareness that the con� icts sometimes arecritical in nature, i.e. what is part of the given order of things or considered ‘good’when view by those operating within one perspective may, in fact, be functioning totake advantage of or even subjugate some when viewed from a different perspective.I � rst became aware of this phenomenon in the process of articulating Question 13.Interestingly, I had dealt with conundrums multiple times before this point, e.g.‘How can I parse knowledge in some way meaningful and useful for me?’ (Q. 11),that were instances of my gaining deeper understanding of some aspect of construc-tivist teaching. The catalyst for Critical Rationality, however, was a con� ict inhuman interest. At this point, I was dealing not merely with an academic problem;I was dealing with an issue having implications of a moral nature.
The initial framing of the question occurs within the taken-for-granted assump-tion that following the students’ interests during instruction in a constructivistclassroom is good. By the end of the question I have come to question thatassumption based on consideration of the teaching–learning transaction from otherperspectives. As a result of thinking about the conditions I posed, I became awarethat differential effects would be likely for various students, depending whoseinterests were followed. I also realized that students’ reactions to the interest-
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Constructing Constructivist Teaching 315
generated excursion would likely vary depending on the social and power relationsexisting among them. The example follows:
Question: If I have introduced a particular topic in class, how can I followthe lead of the students Condition 1: in exploring the content in which theyare interested Condition 2: at they time they are interested in it Condition 3:and also make sure we cover the content they need to know? QuestionElaboration: Part of this question has to do with how to go back after theexploration and, Condition 4: pick up the pieces that have been omitted butwill be needed for subsequent understandings. Condition 5: Another parthas to do with the accumulated effects of spending extra time going backand picking up omitted pieces. The class is designed for K–6 certi� cation,and it’s really easy to spend the whole semester on the early childhoodcontent and neglect the upper elementary content. Con� ict 1: That’s a realdisservice to those students who do not plan to teach in primary grades.Contingency 1: A third piece has to do with the particular interest beingfollowed and how that interest relates to the class as a whole. Con� ict 2:Many students learn best when they are able to connect understandingstogether coherently and only if side excursions are kept to a minimum.Contingency 2: A fourth piece has to do with the number of students whodetermine the direction of the class and whether a small number ofstudents do so, and, if so, Contingency 3: how that affects the rest of thestudents compared to my providing the direction? (Q. 13)
This question, which contains � ve conditions and three contingencies, clearlyreveals depth of understanding of the problematic situation (Dewey, 1933). Inherentin my constraint about picking up pieces in a conceptually sound way is myawareness that elementary mathematics has some logically and conceptually soundstructure. Underlying the constraint about time is my concern that student satisfac-tion with the course will diminish if we do not cover a broad range of content in it.Underlying the constraint about the particular interest being followed is my aware-ness that different social and power relations among students matter, and differentstudents will likely choose different leads, dependent on their epistemology. Con-straints three and four, � rst introduced in the previous question, are particularlyimportant as contemplation on them opened up critical rational thinking I hadotherwise not conceived.
Question Patterns
Analyzing the iterations of the questions, I was able to detect two patterns in thesequence of questions that I deemed worth further exploration. Analysis of onepattern revealed my concern with the beliefs of the students; analysis of the otherrevealed what appeared to be the catalyst for critical rational thinking.
Beliefs and epistemology as motivating concerns. Underlying questions 9–12, 14, and16 are concerns related to the epistemology and the beliefs pre-service teachers hold.
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The concerns expressed in these questions are framed by my own sense-making ofthe interaction between the literature I have read and students’ attitudes andbehaviors I have observed in class. My thinking, particularly related to epistemology,has been informed by the writings of Perry (1968), King and Kitchener (1994),Kuhn (1992), and Hofer and Pintrich (1997). Put very simplistically, my thinkingabout beliefs and epistemology as separate and distinct constructs is that epistemol-ogy is that set of meta-beliefs about the very nature of knowledge and the process bywhich it can be gained that comprises the framework, the perspective, or the lensthrough which beliefs attached to particular types of knowledge gain their parame-ters. For example, I suspect that an epistemology that knowledge is certain andcomes from experts or authorities shapes beliefs about teaching, rendering knowl-edge of content and knowledge of teaching entirely propositional (factual). Thisknowledge is contained in textbooks, Internet, etc. or can be obtained from experts,who, for pre-service teachers more often than not are practicing teachers (White,2000).
In speculating about the nature of the relationship between knowledge and beliefsas I have conceptualized it, I do not suppose that the relationship is one way, i.e. Ido not believe that epistemology gives rise to beliefs but beliefs do not affectepistemology. Said another way, I do not believe that humans are born with thepredisposition to have particular views of the nature of knowledge and the processof knowing. My working hypothesis is that epistemology is abstracted from lifeexperiences with knowledge and knowing. This leads me to believe that epistemol-ogy can be altered and that the medium for doing so is experience. In short, I havecome to posit that the relationship between epistemology and beliefs is reciprocalrather than one way. What appears to be the case, however, is that epistemology iseven less amenable to change than are beliefs, which would be expected if epistemol-ogy exists at a more general or meta-level than beliefs, which are grounded inexperiences. My aim, therefore, is to help my students acquire new understandingsof knowledge and knowing through experiencing these new ways in my class. Myhope is that these new understandings will, in turn, affect their beliefs aboutteaching, and, eventually, their epistemology.
The concern underlying Questions 9 and 10 is how to deal productively with thenotion of teaching as telling. I am aware that pre-service teachers tend to believeteaching happens via didactic methods (Comeaux, 1992) of handing knowledge tostudents. In Question 9, I ask how I can create tasks that require students to cometo justi� able conclusions based on a well-conceived integration of newly gainedinformation with existing knowledge. My aim is to help my students experience theconstruction of understanding that comes from rational sense-making. I want themto see clearly that in mathematics, telling as a teaching method cannot favorablycompare with a problem-solving approach to learning.
In Question 10, I ask whether it is necessary for students to make explicit tothemselves what they have learned in order for transfer to occur to their (eventual)teaching. I am wondering whether the hidden curriculum of my classroom willfunction to transfer to my students’ teaching knowledge the notion that telling andmindless showing are not all right. Because I believe my efforts to help my students
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Constructing Constructivist Teaching 317
gain this understanding are largely futile, I am hoping the hidden curriculum mightwork even if my instruction does not!
In Question 11, my sphere of concern has moved to dispositions and to epistemol-ogy. In it I ask how I can parse all the types of knowledge I want to include in thecurriculum into some typology and how they all meld and actualize in real teachingsituations. I also ask how I can in� uence personal dispositions toward learning. Toexplain the lack of success that spawned Questions 9 and 10, I have extended myarena of concern to the inner depths of the teacher self. I am thinking the curriculumI teach becomes enacted in teaching, depending on each student’s dispositiontoward learning, particularly with respect to openness, curiosity, and questioning(Tishman et al., 1995) and on each student’s epistemology. I further suspect thatthere is a relationship between disposition and epistemology that affects teachingactions.
Because I believe my students will value and encourage in their students thequalities of openness, curiosity, and questioning only if they themselves possessthose same qualities, my concern is with how to help my students develop them. Inaddition, I suspect that whether the students will possess these qualities is in� uencedby their epistemology, particularly with respect to the certainty of knowledge. Thebelief that knowledge is certain is unlikely to engender a disposition toward beingquestioning. Further, the epistemology of college seniors is typically relative, i.e.they believe knowledge regarding ill-structured situations such as those in class-rooms is a matter of opinion. Hence, the probability that they operate out of thebelief system engendered by their upbringing is high. I fear that the upper middle-class white students in this private university, therefore, may frame their teaching interms of their own upbringing, rendering them unable to work productively withstudents who are different from them.
In Questions 13, 14, and 15, I revisit curriculum in terms of mathematics contentand in terms of time to accord to different arenas of curriculum (content, pedagogy,disposition); then I revisit task design. By Question 16, issues related to epistemol-ogy again surface. Question 16 re� ects my concern about the students’ beliefs aboutcontent knowledge. In it I ask, ‘How can I limit what I try to teach so the studentshave deep understanding of core concepts rather than surface understandings of abroad range of ideas?’ Speci� cally, I seek to identify the core understandings ofcontent, pedagogy, and disposition that are foundational and upon which thestudents can build more sophisticated and interrelated understandings.
I am aware that many students believe that content consists of propositions thatcan be practiced (Ball, 1989). They believe they either already know or can get fromteachers’ editions of textbooks adequate knowledge of content (Feiman-Nemser etal., 1989). My own observation endorses those � ndings. Part of my task as amethods teacher, therefore, becomes helping them to develop an awareness thatdeep understanding of many taken-for-granted ideas, e.g. the relationship betweenfractions and decimals, exists. Another part becomes helping the students to acquiredeep understanding of core ideas. To address this concern, I continually work ondeepening and re� ning my own understanding of what the core ideas that comprisethe structure of the discipline of mathematics are. I hope to con� gure my teaching
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so that students at least begin to formulate an understanding of what a structure ofa discipline looks like and to realize the enormous potential it holds for connectedteaching and learning.
Shaping Question 16 is an epistemology-related concern about how to workeffectively with one particular subset of students—those who believe knowledge iscertain and simple rather than sometimes being complex and requiring cognitiveeffort to learn (Schommer, 1990, 1992; Kardash & Scholes, 1995), as do some otherstudents. Because absolutists believe knowledge is simple, is certain, and comesfrom authorities or experts, problems in naturalistic settings are viewed as simpleand as having one right answer. People with this epistemology often display biasedassimilation (Kardash & Miller et al., 1993; Scholes, 1995), i.e. when presented withinformation that discon� rms their beliefs (or should discon� rm them), they rejectthe information outright, often becoming even more � rmly convinced of the ‘cor-rectness’ of their own conceptions.
Observing my own students, I have become aware that some of them becomebored, impatient, or even irritated when I attempt to help them learn concepts indepth. They appear simply to fail to mindfully engage in developing understandingand in projecting implications of this understanding. These students’ responses,written or oral, are typically propositional and super� cial. I am often unable toengage in dialogue with their thinking because they do not display any thoughts towhich I can respond. My choices are typically limited to marking the responseright/wrong, my only alternative being to reiterate a set of directions that is alreadywritten, or to reteach! I interpret this (apparent) failure to engage to epistemologythat is absolutist.
In the long run, I believe I am unable to affect the conceptions of thesestudents—hence my concern about them. They typically learn only at a surfacelevel, which results in poor grades on all tasks that require thoughtful rather thanmemorized responses, and they either become confused over the lack of success oftheir method and then withdraw, or they simply become intractable in maintainingtheir view, sometimes growing vindictive toward me. Either way, they place blamefor any lack of success they experience squarely on me and appear to feel like theclass was worthless to them. On that one thing, we agree.
Catalyst for Critical Rational thinking. The second question sequence pattern Ifound salient is that Critical Rational questions were preceded by questions contain-ing issues that require an informed and carefully considered judgment—as opposedto having answers that felt intellectually like ‘right’ ones. I became aware of issuesinherent in the questions only at the end of Question 12, and only after I had gainedconsiderable understanding of the complexity inherent in the question. My 18-question sequence contains two examples of issue-recognition followed by CriticalRational thinking. The � rst happens in Questions 12 and 13. The second examplehappens in Questions 14, 16, and 18.
Recognition of two issues in Question 12 is followed by the � rst instance ofCritical Rational thinking. The � rst issue is whether I should teach the students whatthey will need to � t comfortably in the culture of the local school systems, i.e. teach
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them the skills they need to appear competent to those with whom they will(eventually) teach, or whether I should teach the students what envisioned bestpractice would be, i.e. teaching as recommended by the Standards (2000). Doingone precludes doing the other, not only because there is insuf� cient time to do bothbut also because, in many instances, they are polar opposites of each other. Thosewho believe in and have developed the ability to teach using the problem-solving andinquiry recommended by the Standards (2000) would not be amenable to using thetransmissive pedagogy dominant in the local school systems and vice versa.
The second issue in Question 12 is who chooses the curriculum of the course—the students or me. My operating assumption is that teaching in a constructivistmanner implies that the direction of lessons will follow the interests of the students,i.e. the students will, in effect, choose the curriculum.
In the instance of both issues, a solution desirable from one view is incompatiblewith a solution desirable from another. Becoming aware of these incompatibilitiesenabled me to arrive at the understanding that various students’ interests, iffollowed, will surely take the class into different directions that will likely bedetermined by not only their current state of understanding but also by theirepistemology and basic beliefs about teaching, learning, students, and content.
Critical Rational thinking became manifested in my concern about which beliefsor interests would provide the direction and how those might relate to the beliefs orinterests of the rest of the class (or to my goals for the class). At this point, Iunderstand that different choices on my part will likely affect different studentsdifferently, with bene� cial effects on some but detrimental effects on others. Thisunderstanding demarked the beginning of Critical Rational thinking.
The second example of issue-recognition followed by Critical Rational thinkingoccurred in Question 14. The � rst issue relates to short-term versus long-term needsof students (a question of having the time to do both). I am aware that developingthe disposition to seek out and re� ectively consider the problems of teaching willrequire devotion of considerable class time to teacher problem-solving that willpreclude teaching at least some of the ‘basic skills’ of teaching. The second issueappears in Question 16. This issue relates to deep understanding rather than surfacelearning. Developing deep understanding will require signi� cant time for explorationof both content and pedagogy (the time question once again) but with an additionaltwist—the epistemology of my students (White, 2000) is such that while some ofthem will bene� t from such activities, others will consider them a waste of time. I amresponsible for working with all my students and know that fruitful work is imposs-ible with a ‘turned off’ student.
Identi� cation and speci� cation of these two issues leads me to the CriticalRational thinking in Question 18. In it I pose the general question, ‘What are themoral issues inherent in what I do?’. An example that explicates my Critical Rationalthinking follows the statement. I wonder whether those students who appear to havea � xed set of beliefs that have remained intractable throughout the semester, e.g.those who steadfastly maintain that ‘telling’ is okay because it worked for them,should be allowed to student teach and (eventually) teach children? The con� ictedends spawning Critical Rational thinking are the students whom I teach but who
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pass my course and the pupils they will eventually teach. I fear that these particularstudents will be as intractable as teachers as they were as students, that they will notbe responsive to the needs of children and that they will not only be ineffective butactually do damage to students who are different from then, either in learning abilityor disability, in ethnicity, in culture. In short, I am terribly burdened by my feelingof responsibility for the negative effects on children that I know can be perpetratedby one in� exible teacher who can see only through his/her own eyes.
In summary, a consistent concern in my questions is how to deal in constructiveways with beliefs some students hold and epistemology framing those beliefs. Ibelieve that both are likely to impede the students’ acquisition of pedagogical andcontent knowledge and their personal growth in both disposition and epistemology.The thinking exhibited in my questions becomes more sophisticated over time,eventually becoming critical in nature. This change is gradual and is accompaniedby an increase in knowledge to inform the question and understanding of thecomplexity inherent in the questions.
Reconstructing Re� ection: discussion
As I prepare to again immerse myself in the ever-� owing current of my teaching, Ineed to glean from my re� ection those meanings that will function as leadinghypotheses (Dewey, 1933) guiding my ongoing re� ection. Two of the conclusions towhich I can come appear particularly noteworthy: (a) my acquisition of construc-tivist teaching has been holistic and piecemeal, and (b) Critical Rational thinkingappears dependent on deep understanding of the complexity inherent in naturalisticsituations. I shall speak to each.
Holistic and Piecemeal Acquisition of Teaching Knowledge
During the process of analyzing these questions, I have become acutely aware of thedegree to which my constructive process has been in� uenced by other pursuits inwhich I have been engaged, several of which I can readily identify. One in� uencewas a research project in which I investigated the epistemology of pre-serviceteachers (White, 2000)—hence my awareness of likely effects of epistemology oncourse experiences. Another in� uence was reading and preparation for teachingother courses, especially one in learning and assessment—hence the awareness ofassessment issues such as match of assessment item to the construct being assessed.A third in� uence was interactions with colleagues, particularly one with whom Iclosely worked for 2 years on a research project involving three teacher educators’thinking and problem-solving during lesson planning. At the same time, I have alsogreatly increased my understanding of seminal concepts in elementary mathematicsand have increased my understanding of college-age students in ways other thanepistemologically. My construction, therefore, has not occurred solely in the contextof my elementary math class, nor has it been a linear process, though it started outin linear fashion. Rather, it has been a dynamic, interactive process that movedforward in piecemeal fashion. Gaining a tidbit of information relevant to one
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question often fomented repercussions in my whole emerging world-view. Using thechanged understanding often made possible the articulation of a condition onanother question or a constraint on another answer. Bits of information came froma variety of sources, and � ashes of insight occurred unexpectedly. Slowly, slowly butsatisfyingly surely, my understanding continues to grow.
Critical Rational Thinking as an Outgrowth of Informed Understanding of the Complexityof Naturalistic Situations
I am poignantly aware that being critical came about approximately 3 years into myconstructive process and only after I had uncovered an issue for which there is no easy or‘one right’ answer. There is a history in teacher education of failure to get pre-serviceteachers to be critical. Rather than question accepted practice, they tend to acceptthe practice, especially of the classroom in which they student teach, withoutquestion (Beyer, 1984; Calderhead, 1987; Zeichner & Liston, 1987). The tendencyis to view classroom phenomena in super� cial ways. Based on my research onmyself, I now have an idea why. This experience leads me to conjecture thatbecoming critical requires a substantial body of knowledge, a disposition to ques-tion, a deeply held af� nity for fairness and objectivity, and a strong enoughself-concept to believe that the only answers worth having are those that possessintegrity by virtue of having been attained through consideration of all availableinformation but without bias toward any view (Baron, 1991; Perkins et al., 1991;Voss, 1991).
Thinking about how to go about this task, I am reminded of the compellinginterest case studies for teacher problem-solving engender in students. I have usedthem as an intrinsic part of an educational psychology class but have not done so inmy elementary math class—too little time and a lack of good teaching problems.Maybe I need to consider their use in a major way in this class. Another tactic Imight use, based on the fact that devising tasks to particular speci� cations openedup avenues of exploration for me, is to ask students to design tasks that accomplisha particular purpose but embed in the purpose some speci� cation that requires astretch for them. Perhaps I can create a series of tasks that require increasedunderstanding of the complexity of learning and knowledge acquisition.
Putting Re� ection and Research on it to Rest—for Today: conclusion
The process of constructing and of researching my re� ection has been personallyvery satisfying to me. Viewed from my current vantage point, it is dif� cult for me toface and acknowledge the naivetey with which I began this venture of teaching inhigher education—after 25 years of K–12 teaching! At the same time, I am awed atthe growth this whole endeavor has enabled me to experience.
I am very aware that my knowledge and experience as a researcher have been ofenormous bene� t. I doubtless would have been able to take some increased under-standing of my process of coming to understand from a surface analysis of the typea non-researcher would likely be able to do—in fact, the type I did on my � rst few
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passes through the data. The most important aspects of what I have learned,however, I would not now know. Having a background in re� ection groundedin the writing of Dewey (1933), Habermas (1974), and Van Manen (1977), aswell as being knowledgeable about the work of teacher educators, particularlyKen Zeichner (Zeichner & Tabachnik, 1982; Zeichner & Liston, 1987; Zeicher &Gore, 1990), provided a framework that became increasingly viable as a lensfor analysis as I made succeeding passes through the data. Because I valuethis particular lens, I also value highly what I have learned about my thinkingabout teaching as a result of analysis. Though I have no doubt that I or othersmight analyze my questions through a different lens and attain other relevantand valuable information about my thinking, I highly recommend the process ofcareful analysis through some lens as an extension to re� ection that is trulymind-making.
At this point, I have multiple questions unanswered or only partially answered andother questions that exist as only a glimmer of emerging insight. For example, thisin-depth analysis has revealed to me the possibility that knowledge may exist in mymind but at a meta-level to which I do not have ready access. If that is the case, thena meta-metacognition is also functioning to guide my questioning process andacquisition of knowledge. Some questions, however, I have answered in ways I canjustify to myself or to any other reasonable person. I am, furthermore, con� dent thatmost of the students whom I teach today do indeed bene� t in many ways from whatI have learned, absolutists notwithstanding. I also am convinced that the mostimportant goal for my teaching is to help my students develop the disposition toquestion even though that is often uncomfortable or even foreign to them. My hopefor them is that they have or develop the vision to pose questions that stretch theirunderstanding, the willingness to seek answers even when the answers are not blackor white or when they run counter to what the pre-service teachers thought theyshould be, and that they have the courage to speak with the integrity born of a senseof fairness and objectivity even when its unpopular to do so. My hope for myself isthat I do the same.
References
BALL, D.L. (1989) Breaking with experience in learning to teach mathematics: what do they bringwith them to teacher education? paper presented at the Annual Meeting of the AmericanEducational Research Association, San Francisco.
BARON, J. (1991) Beliefs about thinking, in: J.F. VOSS, D.N. PERKINS & J.W. SEGAL (Eds) InformalReasoning and Education (Hillsdale, NJ, Lawrence Erlbaum Associates).
BEYER, L.E. (1984) Field experience, ideology, and the development of critical re� ectivity, Journalof Teacher Education 35(3), pp. 36–41.
CALDERHEAD, J. (1987) The quality of re� ection in student teachers’ professional learning,European Journal of Teacher Education 10, pp. 269–278.
COMEAUX, M. (1992) Challenging students’ views about teaching and learning: construc-tivism inthe social foundations classroom, paper presented at the Annual Meeting of the AmericanEducational Research Association, San Francisco, April.
Curriculum and Evaluation Standards (1989) (Reston, VA, National Council of Teachers ofMathematics).
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DEWEY, J. (1933) How We Think (Boston, MA, D.C. Heath).FEIMAN-NEMSER, S., MCDIARMID, G., MELNICK, S. & PARKER, M. (1989) Changing beginning
teachers’ conceptions: a description of an introductory teacher education course (ResearchReport 89–1). (East Lansing, MI, National Center for Research on Teacher Education,College of Education, Michigan State University).
HABERMAS, J. (1974) Theory and Practice (Boston, MA, Beacon Press).HOFER, B.K. & PINTRICH, P.R. (1997) The development of epistemological theories: beliefs about
knowledge and knowing and their relation to learning, Review of Educational Research 67,pp. 88–140.
KARDASH, C.M. & SCHOLES, R.J. (1995, April) Effects of pre-existing beliefs, epistemologicalbeliefs, and need for cognition on interpretation of controversial issues, paper presented atthe Annual Meeting of the American Education Research Association, San Francisco.
KING, P. & KITCHENER, K.S. (1994) Developing Re� ective Judgment (San Francisco, CA, Jossey-Bass).
KUHN, D. (1992) The Skills of Argument (Cambridge, Cambridge University Press).LINCOLN, Y.S. & GUBA, E.G. (1985) Naturalistic Inquiry (Newbury Park, CA, Sage).MILLER, A.G., MCHOSKEY, J.W., BANE, C.M. & DOWD, T.G. (1993) The attitude polarization
phenomenon: role of response measure, attitude extremity, and behavioral consequences ofreported attitude change, Journal of Personality and Social Psychology 64, pp. 561–574.
PERKINS, D., FARADY, M. & BUSHEY, B. (1991) Everyday reasoning and the roots of intelligence,in: J.F. VOSS, D.N. PERKINS & J.W. SEGAL (Eds) Informal Reasoning and Education (Hills-dale, NJ, Lawrence Erlbaum Associates).
PERRY, W.G. (1968) Patterns of development in thought and values of students in a liberal artscollege: a validation of a scheme (Cambridge, MA, Bureau of Study Counsel, HarvardUniversity) (ERIC Document Reproduction Service No. ED 024315).
Principles and Standards for School Mathematics (2000) (Reston, VA, National Council of Teachersof Mathematics).
SCHOMMER, M. (1990) Effects of beliefs about the nature of knowledge on comprehension,Journal of Educational Psychology 76, pp. 248–258.
SCHOMMER, M. (1992) Epistemological development and academic performance among second-ary students, Journal of Educational Psychology 85, pp. 1–6.
TISHMAN, S., PERKINS, D. & JAY, E. (1995) The Thinking Classroom: learning and teaching in aculture of thinking (Boston, MA, Allyn & Bacon).
VAN MANEN, M. (1977) Linking ways of knowing with ways of being practical, Curriculum Inquiry6, pp. 205–228.
VOSS, J.F. (1991) Informal reasoning and international relations, in: J.F. VOSS, D.N. PERKINS &J.W. SEGAL (Eds) Informal Reasoning and Education (Hillsdale, NJ, Lawrence ErlbaumAssociates).
WHITE, B.C. (2000) Preservice teachers’ epistemology viewed through perspectives on problem-atic classroom situations, Journal of Education for Teaching 26, pp. 279–305.
ZEICHNER, K.M. & GORE, J.M. (1990) Teacher socialization, in: W. ROBERT HOUSTON (Ed.)Handbook of Research on Teacher Education.
ZEICHNER, K.M. & LISTON, D. (1987) Teaching student teachers to re� ect, Harvard EducationalReview, 57, pp. 23–48.
ZEICHNER, K.M. & TABACHNICK, B.R. (1982) The belief systems of university supervisors in anelementary student teaching program, Journal of Education for Teaching 8, pp. 35–54.
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Appendix A. Original Questions
1. When should I answer questions, and when should I re� ect them back to the asker?2. What should get constructed? I know it doesn’t make sense to have students try to construct
something procedural like how to use a calculator, but I don’t know what they shouldconstruct. There is a lot of content in this course, and this is my � rst foray through it so Iam unable to determine which ideas might be more or less important than others.
3. How can I ask questions that function to direct the students’ minds in a direction that willprove fruitful, rather than asking questions that have an obvious (one right) answer that don’tincrease their understanding?
4. How can I design tasks that will require construction on the part of the students? I do notknow how to situate my thinking so I can view content as problematic rather than as given.Nor do I know how to design exploratory activities that will enable students to come tounderstand on their own.
5. Is construction different from comprehension, and, if so, how is it different? When studentsread the textbook with comprehension by matching what they read with analogous instancesfrom their own experience, is that construction? Can a student, for example, construct anunderstanding of a lecture? Or, does construction occur only when something has beencreated inside the mind of a person that was not in existence for that person beforehand?
6. How do I deal with the teaching of facts? If content can be parceled out into facts, concepts,and applications, and if concepts are relationships valuable for students to construct, whatabout facts? If learning is to occur, information the student did not know previously has tobe acquired by them. Do the students acquire facts through reading or through videotape?How about through lecture?
7. How do I assess what students construct? If construction, by de� nition, requires that what’slearned be connected to what’s known, and if what’s known is different for each person, howcan I devise assessment tasks that will let me know if the students really do know what I amtrying to assess? What if the students connect the knowledge to something personal asopposed to other mathematical knowledge? What if they interweave what they construct withwhat’s already known so that something new and different has been created? How can I assesstheir grasp of the knowledge I want to assess? Would it ever be possible for me to assigndifferent letter grades to students who have knowledge that is constructed in personal ways,and, if I cannot do so, what is the purpose of formal assessment?
8. How can I sort out the nature of knowledge and the nature of the connecting hook for theindividual? Does some knowledge hook onto pre-existing knowledge, making it slightlypersonal and mostly public, for example, what a common denominator is or how one issupposed to behave at a basketball game? Is other knowledge mostly personal and slightly, ifat all, public, e.g. how one likes to spend spare time or how one typically reacts tocompliments? Can knowledge be separated into the arenas of public and personal, and canI expect to be able to assess the part that is public while leaving the personal intact? Will theconnecting hook always be personal? Does it matter if it is if I only assess the knowledge—notthe hook?
9. How can I create worthwhile tasks that require construction of scienti� c understanding on thepart of the students, assuming that a scienti� c rather than an entirely personal understandingexists? I need to know how to devise tasks that are worth students’ time, that students havethe necessary prior knowledge to complete, and that require them to utilize information fromthe assigned reading or presented in class to arrive at a conclusion that is reasonable andjusti� able. An example of such a task would be one that would make it clear to students whobelieve that ‘telling’ their students how to do math is okay because they were taught that wayand it worked for them that telling is most often inadequate because students memorize howto do exercises for the time being but are unable either to remember how to do them or underwhat circumstances to apply them at a later date.
10. Do students have to make explicit to themselves what they’ve constructed in order for it to
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be encoded in a way that makes it useable for application purposes at a later date? The realquestion here pertains to mathematical knowledge rather than social or personal knowledge,though all three can be acquired by way of the ‘hidden’ curriculum. If the students don’tmake explicit to themselves what they know, will they have incorporated the new knowledgein a scienti� c way or will they still be operating later in terms of their naive understanding?If, for example, the teacher decides at the end of a lesson that the student has the desiredunderstanding, does the student also need to make explicit to him/herself the nature of theunderstanding in order for it to become part of prior knowledge to be utilized in subsequentunderstandings?
11. How can I parse knowledge in some way meaningful and useful to me? I need some way toframe knowledge so that I am able to think about all the kinds I would like to in� uence inpre-service teachers, including that related to content, pedagogy, characteristics of students atdifferent ages and how those interact with pedagogy and content, and how all these meld andget played out in speci� c situations. I also would like to in� uence personal dispositions towardlearning as well as knowledge and beliefs about teaching and learning. Classifying words thatcome to mind are public, personal, facts, concepts, applications, mathematical, social, skills,practical, procedural. I don’t know what set of words, if any, will include and exhaust all thecategories in which I’m interested.
12. Who decides what get constructed? Do I or do the students? If I decide, how do I choose whatcontent to include and what to ignore? Do I base my choice on the discipline of mathematicsand/or the Standards while keeping an eye out for the requirements of the State Board ofEducation for a K–6 class, or do I base it on what students need to know to � t comfortablyinto the school system of this city. I’m concerned that if I choose the content, the studentsmay perform course requirements but discount their value in teaching. They will take out ofthe course only those parts that � t into their belief system about the nature of mathematicsand how it should be taught. On the other hand, if the students decide, they will base theirchoices on their interests or perceived needs. While many of those will surely be worthwhile,some will be super� cial and a poor way to spend valuable class time. Concepts studentschoose may or may not be those needed to understand the structure of arithmetic or toperform the skills they need to be able to match pedagogy to instructional purpose. A tenetof constructivist learning theory is certainly that the interests of the knower are important inthe learning transaction, but balancing students’ interests against the structure of thediscipline makes tough pedagogical terrain for me to negotiate.
13. If I have introduced a particular topic in class, how can I follow the lead of the students inexploring the content in which they are interested at the time they are interested in it and alsomake sure we cover the content they need to know? Part of this question has to do with howto go back after the exploration and, in some conceptually sound way, pick up the pieces thathave been omitted but will be needed for subsequent understandings? Another part has to dowith the accumulated effects of spending extra time going back and picking up omittedpieces. The class is designed for K–6 certi� cation and it’s really easy to spend the wholesemester on the early childhood content and to neglect the upper elementary content. That’sa real disservice to those students who do not plan to teach in primary grades. A third piecehas to do with the particular interest being followed and how that interest relates to the wholeclass. Many student learn best when they are able to connect understandings togethercoherently and only if side excursions are kept to a minimum. A fourth piece has to do withthe number of students who are determining the direction of the class and whether a smallnumber of students are doing so, and, if so, how that is different to the rest of the studentsfrom my providing the direction.
14. What balance do I strike between helping students learn content and pedagogy and helpingstudents acquire (or further develop) the disposition to seek out and re� ectively consider theproblems of teaching? This disposition stands in opposition to one in which students do notsee problems that exist or view problems as simple and as having prescriptive answers. Theproblem here is that dispositional change requires that considerable class time be devoted to
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the analysis of problematic situations rather than learning the knowledge base—whether thatconsists of teaching strategies or knowledge of content—but we already lack time enough inone semester to teach the content or pedagogy. Confounding that issue is that fact that thestudent must possess a knowledge base of content and pedagogy to be functional in studentteaching.
15. How can I devise problems for students to work on when what a problem is may be differentfrom student to student and from group to group? One person or group’s problem may beanother person or groups’ knowledge. Confounding this issue is the fact that level ofeducation (graduate or undergraduate) and degree of experience in the classroom both makea difference. For example, I can initiate a discussion about grading issues and will get almostno response from undergraduates but will get animated discussion from graduates or thosewith experience in the classroom. If I try to point out the issues to undergraduates, thestudents are polite and make an effort to participate but show no real understanding of orinterest in the issue. Because bridging the theory–practice gap is an important part of themission of teacher education, I believe it’s important to prepare students as well as possiblefor issues with which I know they will have to deal. My question becomes two-pronged. Howdo I pose problems of teaching that are real to students of different levels and how do Iinterest students in problems that are real but for which they show no interest?
16. How can I limit what I try to teach so the students have deep understandings of essential ideasrather than surface understandings of a broad range of ideas? Related to that are two issues:(1) what are the essential/core understandings of content, pedagogy, and disposition that arefoundational and upon which the students can build more sophisticated understandings, and(2) how do I work with those student who prefer quick answers and who consider time spentexploring ideas wasted? In fact, the same problem may exist for other students as well becausethey are conditioned to ‘covering’ a lot of material in a semester’s time. They too may not feelthey ‘get their money’s worth.’
17. What proportion of students’ knowing about teaching is public knowledge (that found intextbooks and obtained from professors or other experts), and what part is personal (thatassociated with individual beliefs and experiences)? Can I assess and give grades representingonly public knowledge or can I also give grades representing personal knowledge (includingpersonal growth), and, if so, how can I assess personal growth? I wonder if this question isrelated to the dichotomy in research circles that places personal narratives in opposition tomore traditional ways of knowing.
18. What are the moral issues inherent in what I do? I am able to identify some arenas ofdiscomfort which I suspect are indicators of a personal moral dilemma. For example, I amreally uncomfortable if Elementary Math Methods students who don’t ‘get it’ in terms of amind set in which students have ownership of ideas earn an A in my class. I also am uneasywhen I think about students who appear to have a � xed set of beliefs that remains unchangedduring the semester. I am fearful about how they will interact with a classroom with real kids.I also fret that the time I have allotted for this class is insuf� cient for my students to learnmuch of what I am able to help them learn about math, teaching, and kids. I do not yet haveany of these discomforts translated into moral terms or even know if any one or all of themundergird moral issues. I do know that my work has the potential to affect too many peoplefor it to be moral-free.
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