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(Mis?)Constructing Constructivism
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any educators base recommendati o ns fo r teaching mathemati cs o n "constructi vist" thinking. However, they often mi sunder tand constructi vism, so their recommendations may be incorrect or inappropri ate. We need to examine what constructi vism is and is not, what myths have grown up around constructi vism, and what characteri sti cs define it.
We Do Constructivism on Fridays
Do students stop "constructing know ledge" when their teacher lectures? Do they sw itch over to "absorpti on" mode, pass ively soaki ng in fac ts? o. Constructi vism is not a " type" of learnin g. It does not make sense to be li eve that today a Douglas H. Clements student learns in a constructi vist way, but to mo rrow, in so me di ffe rent way. At its core, constructi vism i a phil oso phy of lea rning th at offers a pe rs pective o n how
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Douglas Clem elliS, ciemellls @acsu.buffalo.edu, teaches at the State Ulli versity oj NelV York at Buffalo, Buffalo, NY 14260. He conducts research ill the areas oj computer applications ill education, early developmelll oj mathematical ideas, alld the learnillg and teaching oj geometry. The allfhor would like 10 thank Mw )' Lindquist, Julie Sarama, Leslie Steffe, and Grayson Wheatley Jor their helpJul commellIS on early drafts oJthis IIwlluscript. FUlldillg Jor this marerial was partially provided by "An Investigation oj the Developl1lellf oj Elemellfary Children's Geometric Thinking ill Computer and Noncomputer Environmenrs," Nar iollal Sciellce Foulldarioll research grallt number ESI-8954664. Any opinions.jilldillgs, alld conclusions or recommelldarions expressed ill this publicarion are those oj the allfhor and do 1I0t necessarily rej1ect the views oJthe National Science Foundatioll. The vielVs expressed in "III My Opillion " do 1I0t necessarily rej1ect the vielVs oJth e EdiIOritd Panel oJTeaching Children Mathemat ics or the National Council oJ Teachers oj Mathematics. Readers are ellcouraged 10 respolld 10 this editorial by selldillg doublespaced leff el's to Teaching Chi ldren Mathemat ics Jor possible publication ill "Readers' Exchange." Man uscripts oj approximately six hundred words are welcomed Jor review Jor " In My Opinioll.'·
people-a ll peop le-learn , all the time. Constructi vism tell s us more about learning than about teaching. However, it has importa nt imp lications fo r teaching. Certain teaching prac tices might be more or less consiste nt with the be liefs of co nstructivism. For exampl e, we could predi ct that we wo uld have less success if we simpl y fed students info rmati o n with no concern fo r connecting that in fo rmation to know ledge they a Lready have.
What Constructivism Is Not Confusio n about what constru ctivism does a nd does no t mea n has e nge nd e red a numbe r of my th . U nqu estio ned , s uc h my th s d ilute a nd po llu te constructi vism.
Myth J: Students should always be actively and reflectively constructing. One powerful way to construct knowledge is through con c iou , reflective construction. As educators, we too seldo m give tudents appropri ate time, tasks, and encouragement to think deepl y and to tal k abo ut mathematical idea . Not all constructions are of that type, however, nor should they be. Our mjnds actively con truct ideas without our " workj ng at it" or even being conscious of it. For example, young children con truct the idea of fl ying an imals, which at ftrst may inc lude everything rurborne, because their minds are acti vely building connections, with little formal "teaching" and even less "effort" on their part. Even when we are consciously working on a problem, we are not full y aware of all that we are learning. There are times fo r many diffe rent ty pes of constructing: time fo r "ex periencing"; fo r " intuitive" learning; fo r learning by listening; fo r practice; and fo r consciou , re fl ecti ve thinking. During all these activities, students construct va luabl e, but di ffe rent kinds of, know ledge. We need to balance these times to meet our goals for students.
TEACHING CHILDREN MATHEMATICS
Myth 2: Manipulatives make learners active. A related myth is that when students are using manipulati ves, then they are "acti vely learning." Manipul ati ves can help stude nts ac ti ve ly construct kn ow ledge (Clements and McMillen 1996); however, teachers can also use manipul ati ves to impose prescribed procedures for routine proble m types. Students then learn to use manipulati ves onl y in a rote manner (Cl ements and McMillen 1996). So teaching with manipul ati ves is not necessaril y " teaching constructi vely." Myth 3: Constructivist learners are lonely voyagers. "Constructing their own knowledge" does not imply that stude nts build their ideas in iso lati on. Rather, the phrase means that one person cannot simply and directly transmit know ledge by means of words into the mind of another person. One can say words with intended meaning, but acti ve listeners have to create their own meanings for the words they hear (Steffe, personal communication). In the words of the poet and philosopher, " If [a teacher] is indeed wise he does not bid yo u enter the house of hi s wisdom, but rather leads yo u to the thresho ld of yo ur own mind" (Gibran 1989). So students do not construct knowledge alone, even though each has to modi fy his or her own ways of thin king and acting. Further, although inventing mathematical ideas together is important, so is learning to better communicate these ideas to others. We must rethink social relations both among students and between student and teacher. For example, constructi vist-oriented teachers must be skilled in structuring the social climate of the classroom so that students discuss, refl ect on, and make sense of mathematics tasks. Myth 4: Cooperative learning is constructivist. Students can work in groups in many differe nt ways, many of which are consistent with constructi vist views of learning. Some approac hes to cooperative learning, however, are based on an absorpti on view. (For a di scuss io n of di ffe re nt approaches, see Nas tas i and Clements [ 199 1].) The way students think and interact is more important than the size of the group in which they work . Just using cooperati ve groups does not necessarily make teaching more "constructi vist." Myth 5: Evely body's right! When we encourage di versity in students' thinking, it is easy to go overboard. After receiving several estimates of the sum of 17 and 2 1, one teacher said, "What's the right answer? Eight is right and thirty-six is right. Everybody's right!" Every student may have valid reasons for his or her solution, but the goal should be building solutions that make sense within the system of mathematics that is socially constructed by the class and the wider community. Thus, everybody's effOtt can be respected without abandoning the notion that some solutions are better than others and that some just do not make sense. DECEMBER 1997
Unique Elements of Constructivism
Intertwined with these myths is the belief that constructi vism is merely a rehas h of past attempts at educati onal reform . Constructi vism is not the onl y view that argues that children learn acti vely or that teachers attempt to understand students' thinking. Constructi vism is part of a di stingui shed intell ectual hi story, including the work of Jean Pi aget and John Dewey, and learning fro m that hi story is important. However, several distingui shing contributi ons of constructi vism set it apart.
Beyond "discovery" Constructivism is not the same as the "di scovery" view promoted earlier in this century that, in one fo rm , advised against telling students anything. Students can construct know ledge, even from lectures, if they li sten to, and think about, what is appropriate for the m. However, a diffe rence ex ists between talking at and talking with students, both in giving students opportuni ties to invent mathematics and in encouraging pos itive beliefs about learning mathe matics. One overarching goal should be that students become autonomous and selfmotivated in their learning. Students should see their "j ob" not as fi ni shing assigned tas ks but as making se nse of, and co mmuni cating abo ut, mathe mati cs (Clements and Batti sta 1990). Constructi vist views pl ace more demands on teachers and children than do "unguided di scovery" approaches.
Beyond "act ive learners" Having students work w ith manipul ati ves and on projects is consiste nt w ith constructivi sm but not unique to it. Co nstructi vist views emphas ize understandin g and buildin g o n stude nts' thinkin g. A lthoug h acco mpli shed in di ffe rent ways, one ma in goal is fo r students to deve lop mathe matical structures that are mo re complex, abstract, and powerful than the o nes they curre ntl y possess (Cle ments and Batti sta 1990; Cobb 1988). No matter how ineffective or inefficient students' ideas and methods mi ght seem, they must be the starting po ints fo r instructi on (S teffe and Cobb 1988). Thi s idea implies that as teac hers we need to learn fro m other teachers and researchers, and stud y our ow n students, to better understand how children thin k. We need to go beyond the general pl atitudes "children are acti ve learners" and "start where the c hild is" when those mere ly mean where he or she is in the tex tbook. Instead, we need to stud y how children think about the parti cul ar mathe mati ca l topics we teach, and we need to work to understand o ur students' thinking at a level deeper than everyday communicati on. To paraphrase Papert (1 980), yo u cann ot learn much about learners unless you learn about learners' learning specific mathe matics.
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Beyond "active teaching" As teachers, we also need to reflect on the developmental progression of children's thinking to understand the wide range of thinking patterns of students in a class and to plan tas ks for groups and individuals. [n so doing, we move beyond traditional teaching and become curriculum builders (Steffe 1991 ). Both teachers and students become more responsible for learning .
Beyond mathematics in books and objects All these ideas culminate in the be lief that mathematics is not " in" textbooks or manipulatives. People create and recreate mathemati cs. Constructivism emphas izes active, living systems, not passive e mpirical data. For constructivi sts, mathematics is the acti vity of constructing patterns and re lationships (Wheatley 1991).
Constructing Constructivism Our dec isions about teaching and learning should emerge from a solid understanding of educational philosophy. Constructivism is a philosophy of learning, not a methodology of teaching. By repudiating myths and understanding its unique elements, we can use constructivism to transform and improve our teaching.
NCTM's
Jobs Online NCTM' s Web site now has a special section for posting "position available" or "position wanted" advertisements for math educators and administrators. Interested in placing an ad? Contact us for the detail s at: • Our Web site- www. nctm .organd see "Jobs Online." • E-mail
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National Counci l of Teachers of Mathemati cs 1906 Association Dr. Reston, VA 20191-1593
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References C lements, Douglas H., and Michael T. Batti sta. "Co nstructi vist Learn ing and Teachin g." AriThmetic Teacher 38 (Septem ber 1990): 34--35. C lements, Douglas H., a nd Sue McMi ll en. " Rethinking ' Concrete' Manipul atives." Teaching Children Math ematics 2 (January 1996) : 270-79. Cobb, Paul. "The Tens ion between Theories of Learning and In struction in Mathematics Ed ucation." Educa tional Psychologist 23 (spring 1988): 87- 103. Gi bran, Kahlil. Th e PropheT. New York: Alfred A. Knopf, 1989. Nastas i, Bonni e K., and Douglas H. Cleme nts. " Research on Cooperative Learnin g: Impli cations for Practice." School Psychology Review 20 ( 199 1) : 11 0-3 1. Papert, Seymour. MindsTOrms: Children , Computers, and Powelfu l Ideas. New York: Bas ic Books, 1980. Steffe, Les lie P. " Mathe matics Curriculum Des ig n: A Constructivist's Perspecti ve." In InrernaTional Perspectives on Trans forming Early Childhood Ma thematics Education, edi ted by Leslie P. Steffe and Terry Wood, 389-98 . Hill sda le, N.J. : Law rence Erlbaum Assoc., 199 1. Steffe, Lesli e P., and Paul Cobb. Construction of A rithmetical Mean ings and STrategies. New York: Springer- Verlag, 1988. Wheatley, Grayson. "Co nstructivist Perspectives on Mathematics and Science Learnin g." Science Education 75 (January 199 1): 9-2 1.
Bibliography Battista, Michael T. , and Dou glas H. C lemen ts. "Constructing Geometric Concepts in Logo." Arithmetic Teacher 38 (November 1990): 15- 17. Bauersfe ld, He inri ch. " Hidden Dime nsions in the So-Call ed Reality of a Mathematics C lassroom." Educational Studies in Math ematics II (February 1980): 23-4 1. Cobb, Paul, Erna Yackel, Terry Wood , Grayson W heat ley, and Graceann Merkel. "C reating a Problem-So lving Atmosphere." Arithmetic Tea cher 36 (September 1988): 46-47 . Davis, Robert B. "G ivi ng Pupil s Tools fo r Thin ki ng." A rithmetic Teacher 38 (January 199 1): 23- 25. Dav is, Robert B. , Caro lyn A. Maher, and Ne l Noddings , eds. Constructivist Views on the Teaching and Learning of Math ematics. l ou mal for Resea rch in Mathematics Education Monograph No.4 . Reston, Va .: National Counc il of Teac hers of Mathematics , 1990. DeVries, Rheta, and Lawrence Kohl berg. Programs of Ea rly Educa tion: Th e Constructi vist View. New York : Longman, 1987. Forman, George E., and Fleet Hill . Constructive Play: Applying Piaget in the Preschool. Rev. ed. Me nl o Park, Calif. : Addi sonWes ley Publi shing Co. , 1984. Kamii , Constance. Young Ch ildren Reinvenr Arithmetic: Implications of Piaget 's Theory. New York: Teachers Coll ege Press, 1985 . - - . " Pl ace Value: An Ex pl anati on of Its Difficulty and Ed ucati ona l Im p li cati o ns for the Primary Grades." l ournal of Resea rch in Childhood Education I (fall/wi nter 1986): 75-86. Papert, Seymour. Th e Children's Ma chine: Rethinking School in the Age of the Compute/: New York: Bas ic Books, 1993 . Steffe , Les li e P. , and Joh n Olive. "The Problem of Fractions in the Eleme ntary School." Arithmetic Teacher 38 (May 199 1) : 22-24. von Glaserfe ld, Ern st. " Learning as a Co nstruc ti ve Act ivity." In Problems of Representation ill the Teaching and Learn ing of Ma thematics, ed ited by C. Ja nvier, 3- 17. Hill sda le , N.J.: Lawre nce Erlbaum Assoc. , 1987. Wheatley, Grayson W. "Calculators and Constructivi sm ." Arithmetic Teacher 38 (October 1990): 22-23. A
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