MARC S. SCHWARTZ KURT W. FISCHER Building vs....

Building vs.

Borrowing

M A R C S . S C H WA RT Z ■ K U RT W. F I S C H E R

The Challenge of Actively Constructing Ideas

Harvard University

SU M M E R 2003 L I B E R A L ED U C A T I O N 23

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THE IMPORTANCE of actively constructing knowl-edge has influenced many generations of ele-mentary and secondary educators, but this con-structivist view rarely extends intopostsecondary classrooms. College educators of-ten harbor the opinion that their students, nowadults, have overcome the need to build theirown knowledge. This assumption is seductivebecause it easily justifies the use of an older,more traditional pedagogy where knowledge istransferred from authoritative sources to readybut naïve recipients. However, the problems gen-erated by this traditional view of education areno less significant at the collegiate level than theyare in elementary and secondary education.

This article explores the tension betweenpassive and active learning among master’s de-gree students working on their teaching cer-tificate in science at the Harvard GraduateSchool of Education. These students wereconvinced that the traditional pedagogy wassufficient until they confronted problems con-cerning floating and sinking that they couldnot solve by listening to lectures or watchingdemonstrations. Our students’ struggle high-lighted the importance of a principle of learn-ing that assumes that building knowledge atany age is an active and step-wise processwhere new knowledge is built upon less com-plex understandings. This hierarchical processsupports all learning and has been explored ina number of theories in cognitive science. We

will introduce one model of development tore-conceptualize the challenge of learning andteaching. Building on this view, we will em-phasize that constructionist pedagogies needto be clearly integrated in university courses(and especially those focusing on teaching).

Constructivism emphasizes that new knowl-edge is a personal creation that is socially me-diated (Tobin and Tip-pins 1993). Universitycourses that overemphasize didacticapproaches risk perpetuating a method ofteaching that is neither useful nor empower-ing; furthermore, they inadvertently under-mine the value of the constructivist approachfor future student teachers and their students.

For five years the first author worked withmore than 100 graduate students in a coursedesigned to explore the role that models playin science.1 Without exception students sharedthe same exasperation with the central mes-sage of the course, that “answers” are only asgood as the models and contexts for whichthey were designed. They regarded this viewof science as unsatisfactory, incomplete, or justanother form of sophism.

Their frustration and story is familiar to sci-ence educators. Our students’ experience wasoften our experience. Their story is embeddedin our story. How then can faculty break thiscycle, and rewrite a story that too often con-cludes with the very best and brightest stu-dents heading towards a trajectory of defeatand disillusionment when facing science as sci-entists, or more generally, when facing the un-knowns in any discipline? How do recent in-sights in cognitive science illustrate howimportant, powerful, and necessary it is for stu-dents to construct their own understanding?

MARC S. SCHWARTZ is educational researcher in thescience education department at the Harvard-Smithsonian Center for Astrophysics. KURT W.

FISCHER is Bigelow Professor of Education anddirector of the Mind, Brain, and Education Programat the Harvard Graduate School of Education.

Constructivismemphasizes that newknowledge is a personalcreation that is socially mediated

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Sally, Rasheed, and Robyn participated in ascience activity exploring the nature of float-ing and sinking. Later they shared theirthoughts about the experience.

Sally: “I didn’t think I would understand, soI didn’t see why I should get involved.”

Rasheed: “I’m not sure we got anything outof this activity. We haven’t been given the an-swers yet.”

Robyn: “I was enjoying the experience oftrying to calculate the density, but my groupasked me why I was doing this. I didn’t knowwhy I was doing it, so I stopped.”

Middle and high school teachers are accus-tomed to these kinds of comments from theirstudents, but they seemed unusual for studentswho had bachelor’s degrees in science. Perhapseven more curious was that our students werealso working on master’s degrees in science ed-ucation as well as state certification to teachscience. One could not have had a brighter,more motivated group of students; however,Sally, Rasheed, and Robyn’s reactions were notunique. They, like their peers, struggled with atopic they believed they had mastered years ago,with a view of science they didn’t understand,and a way of teaching they thought they be-lieved in. Although they could talk articulatelyabout the value of actively constructing knowl-edge, they did not realize how much importancethey had placed on lectures and other strategieswhere they passively received information.Their success in high school and college sciencecourses had been based on evaluations of howwell they recalled this received knowledge, nothow they made sense of this information. Theconstructivist approach had intellectual appeal,but offered little comfort or guidance in theirown learning.

Their consternation began after an activitywhere they compared their answers on twodifferent questionnaires dealing with the na-ture of floating and sinking (Table 1). Afterstudents completed the first questionnaire, theinstructors displayed the distribution of studentanswers for each question, which suggestedthat most students recognized the correct an-swers. No one asked for an explanation. Stu-dents seemed satisfied. We then circulated thesecond questionnaire; however, this time boththe distribution of answers and the students’reactions differed. The distribution of answersappeared to be driven by chance, and thus

24 L I B E R A L ED U C A T I O N SU M M E R 2003

Questionnaire I

1. What is the density of water?A) 0 gm/cc B) 1 gm/cc C) 5 gm/cc

2. If an object’s density is 2 gm/cc, will it float?A) Yes B) No C) Cannot tell from this information

3. If an object has a mass of 2 grams and a volumeof 4 cc, what is its density?A) 8 gm/cc B) 2 gm/ccC) .5 gm/cc

Questionnaire II

1. You come across a canoe in a swimming pool.There is a large anvil in the canoe. You wonder if the level of water in the swimming pool will change if you take out theanvil and throw it in the water. First you closely mark the level of water in the swim-ming pool and then throw the anvil in the water. You notice that: A) The water level goes upB) The water level goes downC) The water level remains unchanged

2. One hot summer afternoon your loved onebrings you a glass of ice water filled to the brim. You look at the glass but do not drink it immediately. It only takes a few minutes for all the ice to melt. After all the ice hasmelted, what happens to the water level? You notice that: A) The water level goes upB) The water level goes downC) The water level remains unchanged

3. Compare two containers filled to the brim with water. One container has a piece of wood floating in it. Which of the following is true?A) The container with just water weighs moreB) The container with the block of wood

weighs moreC) Both containers weigh the same

Table 1: Two questionnaires claiming totest student understanding of densityNote: Each questionnaire contained ten questions thatwere similar in style and content to these questions

students could not determine which answerswere “right.” Even students who answered thequestions correctly were unsure. They imme-diately wanted to know which answers werecorrect. Instead of passing out the answers, thecourse instructors offered students the equip-ment to find the answers. Some students com-mented that they would have preferred theanswers; however, they all were willing to testtheir ideas through experimentation.2 Theywere motivated, self-directed, and successful.They were also unhappy with their effort.Now they wanted to know why the “answer”was the answer.

Their unhappiness highlighted a centralfeature of science—the use of models and thenature of answers. Our students had grownaccustomed to algorithms (e.g., D= M/V orF=ma) and situations where they solved forone or more variables, which the results andreactions to Questionnaire I confirm. Theywere uncomfortable in contexts where formulaswere not useful. They hadn’t thought aboutthe significance of the variables, or recognizedthe relationships embedded in the algorithm’sstructure. As a real world model, the density al-gorithm had limitations, but none that studentshad confronted, as Questionnaire II revealed.

The students and their teachers con-fronted three issues. The first was the needfor new models to handle the second question-naire. The second issue was the meaning ofmodels, including their complex relationshipwith “answers.” But the most important issue,and the one that should concern all educators,is the lack of patience our students displayedin building a personal understanding of themodels they needed to make personal sense ofcomplex phenomena. They preferred to bor-row relationships from their instructors, andexpected that these borrowed relationshipswould provide the insight. Their view con-flicted with the central idea of constructivismthat “learning is a social process of makingsense of experiences in terms of what is al-ready known” (Tobin and Tippins 1993).

Missing modelsModels are conceptual structures scientists andscholars use to tell stories about observed phe-nomena and to predict story lines for those un-investigated parts of the phenomenon. Inclass, our students discovered that some mod-els were more difficult to use than others.

They could verify changes in water level, butcould not explain why the level had changed.They realized that without the appropriatemodel they could not create a compelling storyabout their experiences or with the relation-ships they had discovered.

Given this situation, the course instructorsfelt that the students were ready to explore aversion of Archimedes’ principle: an objectthat floats displaces a volume of water equalto the object’s weight, and an object that sinksdisplaces a volume of liquid equal to the ob-ject’s volume. Some students claimed that theyknew this principle, but only one student in

five years demonstrated any competence usingit. The students asked us to explain how theprinciple worked, which as an exercise in tra-ditional pedagogy we did. We also demon-strated how to generate a solution with one ofthe floating/sinking problems.

We invited our students to continue prac-ticing with the model with our support in anadditional lab period, but they objected,claiming that this work would not be produc-tive. Some argued that the principle did notcount as “an answer” because they stillcouldn’t explain why objects float or sink;however, they no longer asked for explana-tions. They had lost the initiative to explore,


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University ofNebraska–Lincoln

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make sense of a difficult model. Perhapsworse, they no longer had the patience to seethrough a problem even with the offer of timeand support.

Models are often sophisticated mental toolsArchimedes’ principle looks straightforward,and is easy enough to recite; however, fornearly all our students it seemed more like amysterious incantation. Initially they could notuse the principle to explain their observations,much less predict what would happen in unex-plored problems. The principle required stu-dents to coordinate a number of experiencesand concepts: some objects float, some sink; tounderstand why objects sink, one must focuson the object’s volume, however you must shiftyour focus to the object’s weight if it floats; vol-ume is different than weight (which is differ-ent than mass); all three quantities are mea-sured in different ways, with different toolsand are described with dissimilar units. Andthe list is far from complete.

Such coordination requires time and effort,much like learning to juggle. Archimedes’principle is a complex conceptual juggling act.A student must initially learn to juggle asmaller number of the essential experiencesbefore being able to assemble them into perti-nent concepts that can then be coordinatedinto a usable principle. A closer examinationof this learning process reveals a pattern ofevolving complexity in understanding thatmirrors human cognitive development.

Skill theory According to skill theory (Fischer and Bidell1998), people develop toward maturity bybuilding an evolving and hierarchical frame-work of skills over many years, and they usethe same general structure to build new un-derstandings in the short term, working fromlower- to higher-level skills. Learning, inspiredby new problems, involves movement througha portion of one’s developmental framework,and analysis of this process is the basis of astrategy for studying learning called microde-velopment (Wertsch and Stone 1978; Fischerand Granott 1995). Unlike earlier debatesabout whether development to new stages al-lowed the individual to display the new abilityin all contexts (Fischer and Bidell 1998; Ro-

goff 1990), skill theory allows educators toview each new context as an opportunity forstudents to build and coordinate skills usingthe framework of prior skills that is their de-velopmental legacy. However, this process isneither automatic, instantaneous, or guaranteedand often requires extensive construction ofnew lower level skills to create new under-standing. This process frequently requireslengthy time and specific experiences, and it iscontext sensitive (Fischer, et al. 1993; Rogoff1990; Salomon and Perkins 1989).

Long-term development and microdevelop-ment involve movement along the same scalefor growing skills, and Piaget’s framework ofdevelopment provided an important startingpoint for establishing that scale (Piaget 1983).Neo-Piagetian models have focused on charac-terizing the developmental scale behind Piaget’soriginal stages and specifying the processes ofdevelopmental change and construction of un-derstanding (Case 1998; Siegler 1981). Skilltheory (Fischer and Bidell 1998) addressesboth issues. Fischer specifies a scale for devel-opmental change and learning—a series of thir-teen levels defined by specific empirical crite-ria for abrupt or discontinuous (stage-like)change. The levels group into four tiers orstages: (1) Reflexes (to describe developmentin the first couple of months of life), (2) Ac-tions (or Sensorimotor), (3) Representations(or Concrete Operations), and (4) Abstrac-tions (or Formal Operations).

Mastery in each tier progresses through thecoordination of less complex skills into morecomplex ones, with four levels documented ineach tier (Fischer and Bidell 1998). For exam-ple in the representational tier, understandingbegins as “single representations” (e.g., Imaginethat all you knew about the clear liquid in aglass was that people called it “water”. In thiscase the word “water” carries no additionalunderstanding or insight about the liquid in theglass). With further maturity single represen-tations can be coordinated with other singlerepresentations (e.g., Water can be a liquid ora solid). This new level of understanding is amore complex representation, or mapping.Here the observer is able to compare and con-trast object(s) from the perspective of onedimension, e.g., the physical states of water(Parziale and Fischer 1998). Mappings, inturn, can be coordinated with other mappingsabout water or liquids or the objects placed in


water to form a new level—rep-resentational systems. Thetransition to the last tier emergeswhen multiple representationalsystems are successfully coordi-nated together. This fourthlevel becomes the transitionstep to abstract thought, be-cause it forms the first level of the next tier—single abstractions.

The levels specify a scale for not only long-term cognitive development but also short-term learning, including specific processes ofconstructing skills. Researchers and educatorscan use this general framework to model howunderstanding evolves as students confrontnew problems (Parziale and Fischer 1998;Dawson 2002). Students take advantage oftheir cognitive framework to solve problems.Depending on support and context, studentsmove through their framework to more com-plex understandings. Consequently, in shorttime frames (seconds, minutes, hours, or days),students use their cognitive framework in newcontexts and domains to build new under-standings (Fischer, et al. 1993; Granott 1998).

The trajectory of this path oflearning leads to the mostcomplex level that the frame-work will accommodate due toits state of maturation (the op-timal level—the upper limit onskill complexity).

To understand Archimedes’principle, the number of experiences thatneed to be coordinated into single representa-tions which in turn need to be coordinatedinto mappings and then systems is substantial.Even though our master’s students are very ca-pable of creating abstract ideas in areas wherethey have extensive experience, they still needtime to explore, build, and coordinate appro-priate relationships in the “floating and sink-ing” domain in order to take full advantage oftheir developmental legacy. There are no shortcuts to the kinds of abstractions that we wantour students to understand. Attempts at short-cuts lead to rigid and inadequate skills. Al-though our students are capable of buildingtheir own understanding of Archimedes’ prin-ciple, their professors cannot directly givethem their understanding.


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Harvard University

There are no short cuts

to the kinds of abstractions that

we want our students to understand

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The issue that we confronted, and that teachersneed to face, is that we build our understandingsout of our efforts to coordinate our experi-ences. In most traditional educational experi-ences we teachers unconsciously ask studentsto borrow our representations (orArchimedes’ representation) to make furthersense of the world. The complexity ofArchimedes’ representation requires thatteachers let go of the temptation to supplantthe students’ need to construct their own un-derstanding of the concept. This challengedoes not mean that students have to recon-struct Archimedes’ principle on their own with-out guidance. They must make personalmeaning of the principle in a variety of con-texts, guided by teacher and text. Archimedes’model serves as a construct that organizesproperties of the world, but it cannot substitutefor the process of making sense of the modelas students actively put it to use. This operationis an ongoing and dynamic process becausenew problems will challenge the student’s un-derstanding of the principle and the stabilityof this abstraction across contexts.

Thus teachers should resist the instinct toimmediately answer student questions be-cause quick answers create new educationalproblems. The first problem is that providinganswers—a transaction typical in traditionaleducational settings—cannot replace the coor-dination that students need to perform ontheir own. Authoritative interventions can un-dermine the patience necessary for creatingcomplex representations and abstractions, andultimately student confidence that they can dothis kind of work. Our students were too easilysatisfied by ideas that they could repeat back,but had not processed.

Second, quick answers downplay the importantfact that all answers depend on the model instruc-tors emphasize. Why an object floats can be un-derstood from Archimedes’ principle as a specialsituation where the weight of water displaced bythe object equals the weight of the object. But stu-dents can ask why is this so? Newtonian physicsmight address this question from the perspectiveof forces. The force of gravity on the object is bal-anced by the force exerted by the column of waterunder the object. Once again students can askwhy, and once again there is another model (theatomic model) that helps organize necessary obser-vations and representations into a coherent story.

The stories are difficult to tell because if parts ofthe story are changed, then the task of creating analternative conclusion requires a thorough under-standing of the model.

What made questions on Questionnaire Iso popular was that students could borrowand quickly use the density algorithm. Theproblems were designed so that only the lowerlevels of representational understanding wererequired and only recall of the algorithm orsimple facts was needed (e.g., water has a den-sity of 1gm/cc). Students did not need toknow the meaning of this ratio, only that itwas the water’s density (as such, this under-standing illustrates a single representation).The problems in Questionnaire I allowed stu-dents to use the borrowed algorithm to solvefor a missing variable. They did not need toknow what the variables meant or if the con-text mattered as long as they had the vari-ables. Moving past these initial levels into rep-resentational systems and abstractionsrequired more effort. The floating and sink-ing activity wound up irritating our studentsbecause building knowledge beyond the initiallevels was more difficult and time consumingthan borrowed knowledge would accommo-date. In terms of skill theory students wereunable to progress quickly beyond the initiallevels of representations with knowledge thatwas not their own.

Skill theory provides a framework for un-derstanding the cognitive challenge studentsface when confronting new (and abstract)knowledge such as Archimedes’ principle.Constructivism provides a context for sup-porting this unfolding challenge. Tobin andTippit (1993) identify four essential factorsin all variations of constructivism: Newknowledge is a personal construction that issocially mediated. New knowledge is depen-dent on personal experience. New knowledgemust provide reliable and predictable infor-mation about the world. New knowledge isbuilt on existing knowledge. Together bothconstructivism and skill theory help explainwhy Archimedes’ principle can only becomemeaningful to students through their ownconstruction.

In order that students profit from the activeconstruction of knowledge, teachers must fo-cus on several issues in their classrooms. Werecommend including discussions to clarifywhat students believe and why. Any attempt


to amend, enhance, or change student viewsmust account for the experiences that gener-ated their representations. They are intimateand foundational constructions. This founda-tion will help educators identify new, anom-alous, or follow-on experiences that can becomepersonal representations available for furthercoordination as abstractions. This perspectivehelps teachers target with greater precisionthe experiences and discussions that need tooccur if their students are to create the ab-stract concepts demanded by the disciplinethey are studying. Although the capacity ofabstract reasoning is present in adults, thisskill can only emerge when students can cap-ture and coordinate the appropriate sensori-motor and representational understandings.

If teaching at the university level does notemphasize the nature of learning discussed, byembodying its principles, then the universityexperience risks perpetuating a view of learn-ing that only focuses on the manipulation ofborrowed concepts and schemas. Since stu-dents do not build these representations them-selves, the “knowledge” is not long-lived and ismore difficult to coordinate into the kinds ofabstractions that are valued in university dis-course. We invite our colleagues to consider se-riously the need and impact of a constructivistapproach, which requires students to build in-stead of borrowing representations. Tradi-tional teaching practices with sophisticatedstudents can easily mask any deep understand-ing of topics. For colleges of education theproblem is even more insidious in that we riskconvincing another generation of teachers thatthe didactic approach has more value then itactually has. ■■

To respond to this article, [email protected], with the author’s name onthe subject line.

NOTES1 The course was entitled “The Nature of Science” and

taught by three members of the School of Educationand the Faculty of Arts and Sciences: Irwin Shapiro,Bruce Gregory, and Marc Schwartz.

2 Initially, students struggled to produce results everyonecould accept. The instructors were pleased that stu-dents were discovering (or re-discovering) the problemof creating unambiguous experiments. This issue caneasily be reduced to a tedious lecture in methodologyin science courses.

WORKS CITEDCase, R. 1998. The development of conceptual

structures. In D. Kuhn and R. S. Siegler, eds., and W.Damon, series ed. Handbook of child psychology: Vol.2. Cognition, perception, and language. New York: Wiley.

Dawson, T.L. 2002. New tools, new insights: Kohlberg’smoral reasoning stages revisited. International Journalof Behavior Development. 26, 154-166.

Fischer, K.W. and N. Granott. 1995. Beyond one-dimen-sional change: Parallel, concurrent, socially distributedprocess in learning and development. Human Devel-opment, 38, 302-314.

Fischer, K.W. and T.R. Bidell. 1998. Dynamic develop-ment of psychological structures in action and thought.In W. Damon, ed., and R. Lerner, series ed. Handbookof child psychology, Vol. 5, Theoretical models of hu-man development. 467-561. New York: Wiley.

Fischer, K.W., D. Bullock, E.J. Rotenberg, and P. Raya.1993. The dynamics of competence: How context con-tributes directly to skill. In R. Wozniak and K.W. Fis-cher, eds. Development in context: Acting and thinking in specific environments, 93-117.Hillsdale, N.J.: Erlbaum.

Granott, N. 1998. We learn, therefore we develop: Learn-ing versus development—or developing learning. In C.Smith and T. Pourchot, eds. Adult learning and devel-opment: Perspectives from education psychology,15-34.Mahwah, NJ: Lawrence Erlbaum Associates.

Parziale, J. And K.W. Fischer.1998. The practical use ofskill theory in classrooms. In R. J. Sternberg and W.M. Williams, eds. Intelligence, instruction and assess-ment, 96-110. Hillsdale, N.J.: Lawrence Erlbaum As-sociates.

Piaget, J. 1983. Piaget’s theory. In P. M. Mussen, ed.,Handbook of child psychology, Vol. 1, 103-128. NewYork: Wiley.

Rogoff, B. 1990. Apprenticeship in thinking. New York:Oxford University Press.

Salomon, G. and D.N. Perkins.1989. Rocky roads totransfer: Rethinking mechanisms of a neglected phe-nomenon. Educational Psychologist, 24: 2, 113-142.

Siegler, R.S. 1981. Developmental sequences within andbetween concepts. Monographs of the Society of Re-search in Child Development, 46(2, Serial No. 189).

Tobin, K. and D. Tippins. 1993. Constructivism as a ref-erent for teaching and learning. In K. Tobin, ed. Thepractice of constructivism in science education, 3-21.Hillsdale, NJ: Lawrence Erlbaum Associates.

Wertsch, J.V. and C.A. Stone. 1978. Microgenesis as atool for developmental analysis. Quarterly Newsletterof the Laboratory of Comparative Human Cognition,1:1, 8-10.


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