NATIONAL FORUM OF TEACHER EDUCATION JOURNAL VOLUME 21, NUMBER 3, 2011 Critical Thinking and Constructivism Techniques for Improving Student Achievement Fred C. Lunenburg Sam Houston State University ______________________________________________________________________________ ABSTRACT NAEP data suggest that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. Moreover, students in some other advanced, technological countries consistently outperform American students on international tests in science and mathematics. The ultimate goal of the No Child Left Behind legislation is that all students will demonstrate competency over challenging subject matter in the core subject areas— reading, mathematics, science, and social studies—and learn to use their minds well, so they are prepared for responsible citizenship, further learning, and productive employment in our Nation’s economy. In this article, I discuss the condition of education in America and offer two approaches to teaching subject matter (critical thinking and constructivism) that may result in major improvements in student achievement. ______________________________________________________________________________ Accountability for school improvement is a central theme of federal and state polices. The No Child Left Behind Act of 2001 (Public Law 107-110) sets demanding accountability standards for schools, school districts, and states, including new state testing requirements designed to improve education. For example, the law requires that states develop both content standards in reading and mathematics and tests that are linked to the standards for grades 3 through 8, with science standards and assessments to follow. States must identify adequate yearly progress (AYP) objectives and disaggregate test results for all students and subgroups of students based on socioeconomic status, race/ethnicity, English language proficiency, and disability. Moreover, the law mandates that 100 percent of students must score at the proficient level on state tests by 2014. Will schools, school districts, and states be able to respond to the demand? Where Are We Now? The National Assessment of Educational Progress (NAEP), often referred to as the nation’s ―report card,‖ is the only nationally representative continuing assessment that measures what students know and are able to do in the core subject areas. NAEP is administered at fourth grade, eighth grade, and twelfth grade at various points in time. Both public and private 1 NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 2____________________________________________________________________________________________ school students in grades 4, 8, and 12 are sampled and assessed on a regular basis. The NAEP tests are developed nationally by teachers, curriculum experts, and the public. The NAEP is authorized by Congress and directed by the National Center for Education Statistics of the U.S. Department of Education. The data suggests that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. NAEP reports that only one third of 12th graders are able to perform rigorous reading passages. The average reading levels of black 17-year olds is about 4 years behind that of white students and mathematics scores of this group is about 2 years behind white students (Howard, 2011; U.S. Department of Education, 2010a; Paige, 2011). Differences between white and Hispanic reading scores on the NAEP have been declining consistently since 1975 (U.S. Department of Education, 2010a). The gap between white and Hispanic mathematics scores on NAEP has been declining since 1975, as well (U.S. Department of Education, 2010a). Merely 11% of secondary students demonstrate a good understanding of history. The general standards of American schools compare unfavorably with those of other industrialized nations (U.S. Department of Education, 2010b). NAEP data and International Educational Achievement (IEA) studies suggest that students are not learning how to think. In other words, although student learning of facts and basic skills has improved slightly over the past three decades, the development of more advanced reasoning abilities has declined. To achieve major improvements in student achievement will require fundamental changes in the way subject matter is taught. Classroom teachers at all levels should consider critical thinking and constructivism that offer real promise for improving the achievement of all students in the core subject areas. Critical Thinking The concept of critical thinking may be one of the most significant trends in education relative to the dynamic relationship between how teachers teach and how students learn (Mason, 2010). Critical thinking shifts classroom design from a model that largely ignores thinking to one that renders it pervasive and necessary (Cohen, 2010; Tittle, 2010; Vaughn, 2009). Critical teaching views content as something alive only in minds, as modes of thinking driven by questions, as existing in textbooks only to be regenerated in the minds of students. Once we understand content as inseparable from the thinking that generates, organizes, analyzes, synthesizes, evaluates, and transforms it, we recognize that content cannot in principle ever be “completed” because thinking is never completed. To understand content, therefore, is to understand its implications. But to understand its implications one must understand that those implications in turn have further implications, and hence must be explored thoughtfully. The problem with didactic teaching is that content is inadvertently treated as static, as virtually “dead”. Content is treated as something to be mimicked, to be repeated back, to be parroted. And since students only rarely process content deeply when they play the role of passive listeners in lecture-centered instruction, little is learned in the long term. Furthermore, because students are taught content in a way that renders them unlikely to think it through, their minds retreat into rote memorization, abandoning any attempt to grasp the logic of what they are committing to memory. Those who teach critically emphasize that only those who can “think” through content FRED C. LUNENBURG ____________________________________________________________________________________________3 truly learn it (Numrich, 2010). Content “dies” when one tries to mechanically learn it. Content has to take root in the thinking of students and, when properly learned, transforms the way they think. Hence, when students study a subject in a “critical” way, they take possession of a new mode to thinking which, so internalized, generates new thoughts, understandings, and beliefs. Their thinking, now driven by a set of new questions, becomes an instrument of insight and a new point of view. History texts become, in the minds of students thinking critically, a stimulus to historical thinking. Geography texts are internalized as geographical thinking. Mathematical content is transformed into mathematical thinking. As a result of being taught to think critically, students study biology and become biological thinkers. They study sociology and begin to notice the permissions, injunctions, and taboos of the groups in which they participate. They study literature and begin to notice the way in which all humans tend to define their lives in the stories they tell. They study economics and begin to notice how much of their behavior is intertwined with economic forces and needs. There are ways, indeed almost an unlimited number, to stimulate critical thinking at every educational level and in every teaching setting (Dunn, 2010; hooks, 2009; Liecester, 2010). When considering technology for this stimulation, the World Wide Web (WWW) is important to instructional design; it contains three keys to educational value: hypertext, the delivery of multimedia, and true interactivity (Stewart, 2010). These values are operant and alive in the classroom through such applications as: graphics, sound, and video which bring to life world events, museum tours, library visits, world visits, and up-to-date weather maps (Griffin, 2010). Through these WWW mechanisms, a constructivist instructional model advance higher level instruction, such as problem solving and increased learner control. The WWW becomes a necessary tool for student-centered discovery and research. Of course, it can also be used for lower level drill and practice. At every level and in all subjects, students need to learn how to: precisely put questions, define contexts and purposes, pursue relevant information, analyze key concepts, derive sound inferences, generate good reasons, recognize questionable assumptions, trace important implications, and think empathically within different points of view (Dunn, 2010; hooks, 2010; Leicester, 2010). The WWW enables learners and teachers in each area by providing information for good reasoners to figure things out (Bowell; Levy, 2010). Critical thinking may be a key organizing concept for all educational reform (Bulach, Lunenburg, & Potter, 2012). Constructivism Constructivism is another, somewhat related, trend in education that can play a dynamic role in the relationship between how teachers teach and how children learn. One foundational premise of constructivism is that children actively construct their knowledge, rather than simply absorbing ideas spoken to them by teachers (Fosnot, 2006; Phillips, 2000; Larochelle, 2010). For example, Jean Piaget (1970) proposed that children make sense in ways very different from adults, and that they learn through the process of trying to make things happen, trying to manipulate their environment. Theories like these, which assert that “people are not recorders of information, but builders of knowledge structures,” have been grouped under the heading of constructivism (Pass, 2005; Wadsworth, 2004). Thus, students are ultimately responsible for NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 4____________________________________________________________________________________________ their own learning within a learning atmosphere in which teachers value student thinking, initiate lessons that foster cooperative learning, provide opportunities for students to be exposed to interdisciplinary curriculum, structure learning around primary concepts, and facilitate authentic assessment of student understanding. In constructivist theory, it is assumed that learners have to construct their own knowledge—individually and collectively. Each learner has a repertoire of conceptions and skills with which she or he must construct knowledge to solve problems presented by the environment. The role of the teacher and other learners is to provide the setting, pose the challenges, and offer the support that will encourage cognitive construction (Chaille, 2008). Since students lack the experience of experts in the field, teachers bear a great responsibility for guiding student activity, modeling behavior, and providing examples that will transform student group discussions into meaningful communication about subject matter (Flynn, 2005). Constructivism emphasizes the processes by which children create and develop their ideas. Applications lie in creating curricula that not only match but also challenge children’s understanding, fostering further growth and development of the mind (Baltes, 2007; Kincheloe, 2006; Leitner, 2010). Furthermore, when children collaborate in cooperative learning groups, they share the process of constructing their ideas with others. This collective effort provides the opportunity for children to reflect on and elaborate not only their own ideas but also those of their peers as well. With the improvement and access to the WWW, the children’s cooperative classroom becomes the world (Payne, 2010; Stewart, 2010). In this cooperative learning setting, children view their peers as resources rather than as competitors. A feeling of teamwork ensues. These processes have resulted in substantial advances in student learning (Bulach, Lunenburg, & Potter, 2012; Larochelle, 2010; Phillips, 2000). Constructivism is serving as the basis for many of the current reforms in several subject matter disciplines. The National Council of Teachers of Mathematics (NCTM) has published its document, Curriculum and Evaluation Standards for School Mathematics, which calls for mathematics classrooms where problem solving, concept development, and the construction of learner-generated solutions and algorithms are stressed rather than drill and practice on correct procedures and facts to get “the right” answer. The National Committee on Science Education Standards and Assessment similarly has issued its document, National Science Education Standards which calls for science education reform based on experimentation and learnergenerated inquiry, investigations, hypotheses, and models. The National Council of Teachers of English (NCTE) has called for emergent literacy as an important thrust in language arts reform. Interdisciplinary curricula is the theme of social studies reform being advocated by the National Council of Social Studies. Principles of Constructivist Pedagogy Jacqueline Brooks and Martin Brooks provide a detailed description of constructivist classroom practice and its theoretical underpinnings in their book, In Search for Understanding: The Case for Constructivist Classrooms (2005). They provide five principles of constructivist pedagogy: (a) posing problems of emerging relevance to learners; (b) structuring learning around “big ideas” or primary concepts; (c) seeking and valuing students’ points of view; (d) adapting curriculum to address students’ suppositions; and (e) assessing student learning in the context of teaching. FRED C. LUNENBURG ____________________________________________________________________________________________5 Principle 1: Posing problems of emerging relevance to students. Relevance does not have to be pre-existing for the student. Not all students come to the classroom interested in learning. Relevance can emerge through teacher mediation. Principle 2: Structuring learning around primary concepts. When designing curriculum, constructivist teachers organize information around conceptual clusters of problems, questions, and discrepant situations, because students are most engaged when problems and ideas are presented holistically rather than in separate, isolated parts. Much of traditional education breaks wholes into parts and then focuses separately on each part. But many students are unable to build concepts and skills from parts to wholes. Principle 3: Seeking and valuing students’ points of view. Students’ points of view are avenues into their reasoning. Awareness of students’ points of view help teachers challenge students, making school experiences both contextual and meaningful. Teachers who operate without awareness of their students’ points of view often doom students to dull, irrelevant experiences, and even failure. Principle 4: Adapting curriculum to address students’ suppositions. Teacher mediation is a key factor in adapting curriculum to address students’ suppositions. The teacher can abstract student learning or help build their own bridges from present understandings to new, more complex understandings. If suppositions are not explicitly addressed, most students will find lessons devoid of meaning, regardless of how charismatic the teacher or attractive the materials used. While it is the teacher who structures the opportunity, it is the students’ own reflective abstractions that create the new understanding. Principle 5: Assessing student learning in the context of teaching. Multiple-choice, norm-referenced tests are structured to determine whether students know information related to a particular body of knowledge. The overarching question posed by such activities is: What do you know?” Authentic assessment focuses on analytical thinking and performance, whereas norm-referenced, standardized tests focus on low-level rote skills. Becoming a Constructivist Teacher Brooks and Brooks (2005) provide the following set of descriptors of constructivist’ teaching behaviors, which they feel teachers can use to experiment with the approach. The set of descriptors describes teachers as facilitators of learning and empowerers of students to construct their own understandings of content, not simply as providers of information and managers of behavior. Constructivist teachers encourage and accept student autonomy and initiative. Autonomy and initiative cause students' pursuit of connections among concepts. Students who formulate questions and then go on to answer and analyze them are taking responsibility for their own learning and become problem solvers as well as problem finders. NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 6____________________________________________________________________________________________ Constructivist teachers use raw data and primary sources, along with manipulatives and interactive and physical materials. In the constructivist approach to teaching, learning becomes the result of research related to real problems. For example, students can be assigned to read historical accounts of the effects of social policies of the early 1980's on the economic profile of the African-American population in America. Or students can be taught to read the census reports and encouraged to generate their own inferences about social policies. The latter approach allows students to construct their own understandings of the issues. When framing tasks, constructivist teachers use cognitive terminology such as "classify", "analyze", "predict", and "create". Formulating tasks around cognitive activities such as analysis, interpretation, classification, and prediction, and explicitly using those terms with students, fosters the construction of new understandings about content. Constructivist teachers allow student responses to drive lessons, shift instructional strategies, and alter content. This does not mean that students' interest or lack of interest in a topic determines whether the topic is taught or that whole sections of the curriculum will be eliminated. It does mean that constructivist teachers will capitalize on "teachable moments" throughout the school year. These are moments when the students' interest, knowledge, and enthusiasm intersect and transcend a particular lesson. For example, the Persian Gulf War may have provoked student initiated discussion during that time period. Constructivist teachers inquire about students’ understandings of concepts before sharing their own understandings of those concepts. When teachers share their ideas before students have an opportunity to formulate their own, students' examination of their own ideas is eliminated. In such environments, most students will stop thinking about the concept and wait for the teacher to provide the "correct answer". Consequently, students are prevented from constructing their own ideas and theories. Constructivist teachers encourage students to engage in dialogue, both with the teacher and with one another. One way that students change or reinforce their ideas and theories is through social discourse. Students are empowered when they have an opportunity to present their own ideas and hear and reflect on the ideas of others. This process helps students construct new understandings or reflect on their existing ones. According to Robert Slavin (2009), student-to-student dialogue is the foundation upon which cooperative learning is based. Constructivist teachers encourage students’ inquiry by asking thoughtful, openended questions and encouraging students to ask questions of each other. Complex, thoughtful questions, that have more than one response, challenge students to delve into issues deeply and broadly and to form their own understandings of events and phenomena. Constructivist teachers seek elaboration of students’ initial response. Students' initial responses about issues are not necessarily their final thoughts, nor their best thoughts on a topic. Through elaboration of students’ initial responses, they frequently reconceptualize and assess their own errors and, in the process, construct their own understandings of issues, concepts, and theories. FRED C. LUNENBURG ____________________________________________________________________________________________7 Constructivist teachers engage students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion. Cognitive growth occurs when an individual reformulates a current perspective. Students at all levels formulate and refine ideas about phenomena and then tenaciously hold onto these ideas as eternal truths. Even when confronted with authoritative evidence that challenge their views, students generally adhere to their original ideas. When teachers provide experiences that might engender contradictions, the framework for students' original ideas weaken, causing them to rethink their perspectives and formulate new understandings. Constructivist teachers allow wait time after posing questions. In most classrooms, there are some students who are not prepared to respond to questions or other stimuli immediately. They require more time to process information. Teachers that require immediate responses prevent these students from thinking through theories and concepts thoroughly, forcing them to become spectators. These students learn quickly that there is no point in mentally engaging in teacher-posed questions. Constructivist teachers provide time for students to construct relationships and create metaphors. Constructivist teachers structure and mediate classroom activities and provide the necessary time and materials for learning to occur, which causes students to construct patterns, relationships among concepts and theories for themselves. Constructivist teachers also encourage the use of metaphor as a way to facilitate learning. Metaphors help students to understand complex issues in a holistic way and to ruminate mentally with the parts of the whole to determine whether the metaphor works. Constructivist teachers nurture students' natural curiosity through frequent use of the learning cycle model. The learning cycle model has been used in science education for some time (Buxton, 2011). The model describes curriculum development and instruction as a threestep cycle: discovery, concept introduction, and concept application. First, the teacher provides an open-ended opportunity for students to interact with purposefully selected materials. This step is designed to generate student questions and hypotheses from working with the materials (discovery). Next, the teacher provides lessons aimed at focusing the students' questions, providing related and new vocabulary, framing with students their laboratory experiences, and such (concept introduction). Finally, students engage in one or more interactions of the discovery-concept introduction sequence. Students work on new problems with the potential for evoking a reflective, new look at the concepts studied previously (concept application). The aforementioned descriptors of constructivist teaching highlight practices that help students to construct their own understandings of challenging subject matter content. These descriptors can serve as guidelines for interpreting what it means to become a constructivist teacher. For specific examples of how to implement each of the descriptors, see Brooks and Brooks (2005). NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 8____________________________________________________________________________________________ Conclusion NAEP data suggest that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. Moreover, students in some other advanced, technological countries consistently outperform American students on international tests in science and mathematics. The ultimate goal of the No Child Left Behind legislation is that all students will demonstrate competency over challenging subject matter in the core subject areas— reading, mathematics, science, and social studies—and learn to use their minds well, so they are prepared for responsible citizenship, further learning, and productive employment in our Nation’s economy. Critical thinking and constructivism offer real promise for improving the achievement of all students in the core subject areas. References Baltes, P.B. (2007). Lifespan development and the brain: The perspective of biocultural and co-constructivism. West Nyack, NY: Cambridge University Press. Bowell, T. (2010). Critical thinking: A concise guide. New York, NY: Routledge. Brooks, J. G., & Brooks, M. (2005). In search of understanding: The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development. Bulach, C.R., Lunenburg, F.C., & Potter, L. (2012). Creating a culture for highperforming schools: A comprehensive approach to school reform. Lanham, MD: Rowman & Littlefield. Buxton, C. A. (2011). Teaching science in elementary and middle school: A cognitive and cultural approach. Thousand Oaks, CA: Sage. Chaille, C. (2008). Constructivism across the curriculum: Big ideas as inspiration. Upper Saddle River, NJ: Allyn & Bacon. Cohen, E.D. (2010). Critical thinking. Lanham, MD: Rowman & Littlefield. Dunn, D.S. (2010). Teaching critical thinking: A handbook of best practices. New York: Wiley. Flynn, P. (2005). Applying standards-based constructivism: A two-step guide for motivating elementary students. Larchmont, NY: Eye on Education. Fosnot, C.T. (2006). Constructivism: Theory, perspectives, and practice. New York, NY: Teachers College Press. Griffin, R.E. (2010). Engaging creativity and critical thinking. Washington, DC: International Visual Literacy Association. Hooks, b. (2009). Teaching critical thinking. New York, NY: Routledge. Howard, T. G. (2011). Why race and culture matter in schools: Closing the achievement gap in America’s classrooms. New York: Teachers College Press. Kincheloe, J.L. (2006). Critical constructivism primer. New York, NY: Peter Lang. Larochelle, M. (2010). Constructivism and education. West Nyack, NY: Cambridge University Press. Leicester, M. (2010). Teaching critical thinking skills. London, England: Continuum International Publishing Group. FRED C. LUNENBURG ____________________________________________________________________________________________9 Leitner, L.M. (2010). Personal constructivism: Theory and applications. New York, NY: Pace University Press. Levy, D.A. (2010). Tools of critical thinking. Long Grove, IL: Waveland Press. Mason, M. (2010). Critical thinking and learning. New York, NY: Wiley. Moore, B. (2010). Critical thinking and formative assessments: Increasing the rigor in your classroom. Larchmont, NY: Eye on Education. Numrich, C. (2010). Raise the issues: An integrated approach to critical thinking. Upper Saddle River, NJ: Pearson. Paige, R. (2011). The black-white achievement gap: Why closing it is the greatest civil rights issue of our time. New York, NY: Amacom. Pass, Susan (2005). Parallel paths to constructivism: Jean Piaget and Lev Vygotsky. Charlotte, NC: Information Age. Payne, C.R. (2010). Information technology and constructivism in higher education: Progressive learning frameworks. Hersey, PA: IGI Global. Phillips, D.C. (2000). Constructivism in education, Vol. 1. Chicago, IL: University of Chicago Press. Piaget, J. (1970). Piaget’s theory. In P. Mussen (Ed.), Carmichael’s manual of child psychology (Vol. I, pp.703-732). New York: John Wiley. Slavin, R.E., & Madden, N.A. (2009). One million children: Success for all. Thousand Oaks, CA: Corwin Press. Stewart, C.M. (2010). Teaching and learning with technology: Beyond constructivism. New York, NY: Taylor & Francis. Tittle, P. (2010). Critical thinking: An appeal to reason. New York, NY: Taylor & Francis. U.S. Department of Education, National Center for Education Statistics. (2010a). NAEP assessments of fourth, eighth, and twelfth graders. Washington, DC: U.S. Government Printing Office. U.S. Department of Education. (2010b). The condition of education. Washington, DC: U. S. Government Printing Office. Vaughn, L. (2009). The power of critical thinking: Effective reasoning about ordinary and extraordinary claims. New York, NY: Oxford University Press. Wadsworth, B.J. (2004). Piaget’s theory of cognitive and affective development: Foundations of constructivism. Upper Saddle River, NJ: Allyn & Bacon. NATIONAL FORUM OF TEACHER EDUCATION JOURNAL VOLUME 21, NUMBER 3, 2011 Critical Thinking and Constructivism Techniques for Improving Student Achievement Fred C. Lunenburg Sam Houston State University ______________________________________________________________________________ ABSTRACT NAEP data suggest that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. Moreover, students in some other advanced, technological countries consistently outperform American students on international tests in science and mathematics. The ultimate goal of the No Child Left Behind legislation is that all students will demonstrate competency over challenging subject matter in the core subject areas— reading, mathematics, science, and social studies—and learn to use their minds well, so they are prepared for responsible citizenship, further learning, and productive employment in our Nation’s economy. In this article, I discuss the condition of education in America and offer two approaches to teaching subject matter (critical thinking and constructivism) that may result in major improvements in student achievement. ______________________________________________________________________________ Accountability for school improvement is a central theme of federal and state polices. The No Child Left Behind Act of 2001 (Public Law 107-110) sets demanding accountability standards for schools, school districts, and states, including new state testing requirements designed to improve education. For example, the law requires that states develop both content standards in reading and mathematics and tests that are linked to the standards for grades 3 through 8, with science standards and assessments to follow. States must identify adequate yearly progress (AYP) objectives and disaggregate test results for all students and subgroups of students based on socioeconomic status, race/ethnicity, English language proficiency, and disability. Moreover, the law mandates that 100 percent of students must score at the proficient level on state tests by 2014. Will schools, school districts, and states be able to respond to the demand? Where Are We Now? The National Assessment of Educational Progress (NAEP), often referred to as the nation’s ―report card,‖ is the only nationally representative continuing assessment that measures what students know and are able to do in the core subject areas. NAEP is administered at fourth grade, eighth grade, and twelfth grade at various points in time. Both public and private 1 NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 2____________________________________________________________________________________________ school students in grades 4, 8, and 12 are sampled and assessed on a regular basis. The NAEP tests are developed nationally by teachers, curriculum experts, and the public. The NAEP is authorized by Congress and directed by the National Center for Education Statistics of the U.S. Department of Education. The data suggests that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. NAEP reports that only one third of 12th graders are able to perform rigorous reading passages. The average reading levels of black 17-year olds is about 4 years behind that of white students and mathematics scores of this group is about 2 years behind white students (Howard, 2011; U.S. Department of Education, 2010a; Paige, 2011). Differences between white and Hispanic reading scores on the NAEP have been declining consistently since 1975 (U.S. Department of Education, 2010a). The gap between white and Hispanic mathematics scores on NAEP has been declining since 1975, as well (U.S. Department of Education, 2010a). Merely 11% of secondary students demonstrate a good understanding of history. The general standards of American schools compare unfavorably with those of other industrialized nations (U.S. Department of Education, 2010b). NAEP data and International Educational Achievement (IEA) studies suggest that students are not learning how to think. In other words, although student learning of facts and basic skills has improved slightly over the past three decades, the development of more advanced reasoning abilities has declined. To achieve major improvements in student achievement will require fundamental changes in the way subject matter is taught. Classroom teachers at all levels should consider critical thinking and constructivism that offer real promise for improving the achievement of all students in the core subject areas. Critical Thinking The concept of critical thinking may be one of the most significant trends in education relative to the dynamic relationship between how teachers teach and how students learn (Mason, 2010). Critical thinking shifts classroom design from a model that largely ignores thinking to one that renders it pervasive and necessary (Cohen, 2010; Tittle, 2010; Vaughn, 2009). Critical teaching views content as something alive only in minds, as modes of thinking driven by questions, as existing in textbooks only to be regenerated in the minds of students. Once we understand content as inseparable from the thinking that generates, organizes, analyzes, synthesizes, evaluates, and transforms it, we recognize that content cannot in principle ever be “completed” because thinking is never completed. To understand content, therefore, is to understand its implications. But to understand its implications one must understand that those implications in turn have further implications, and hence must be explored thoughtfully. The problem with didactic teaching is that content is inadvertently treated as static, as virtually “dead”. Content is treated as something to be mimicked, to be repeated back, to be parroted. And since students only rarely process content deeply when they play the role of passive listeners in lecture-centered instruction, little is learned in the long term. Furthermore, because students are taught content in a way that renders them unlikely to think it through, their minds retreat into rote memorization, abandoning any attempt to grasp the logic of what they are committing to memory. Those who teach critically emphasize that only those who can “think” through content FRED C. LUNENBURG ____________________________________________________________________________________________3 truly learn it (Numrich, 2010). Content “dies” when one tries to mechanically learn it. Content has to take root in the thinking of students and, when properly learned, transforms the way they think. Hence, when students study a subject in a “critical” way, they take possession of a new mode to thinking which, so internalized, generates new thoughts, understandings, and beliefs. Their thinking, now driven by a set of new questions, becomes an instrument of insight and a new point of view. History texts become, in the minds of students thinking critically, a stimulus to historical thinking. Geography texts are internalized as geographical thinking. Mathematical content is transformed into mathematical thinking. As a result of being taught to think critically, students study biology and become biological thinkers. They study sociology and begin to notice the permissions, injunctions, and taboos of the groups in which they participate. They study literature and begin to notice the way in which all humans tend to define their lives in the stories they tell. They study economics and begin to notice how much of their behavior is intertwined with economic forces and needs. There are ways, indeed almost an unlimited number, to stimulate critical thinking at every educational level and in every teaching setting (Dunn, 2010; hooks, 2009; Liecester, 2010). When considering technology for this stimulation, the World Wide Web (WWW) is important to instructional design; it contains three keys to educational value: hypertext, the delivery of multimedia, and true interactivity (Stewart, 2010). These values are operant and alive in the classroom through such applications as: graphics, sound, and video which bring to life world events, museum tours, library visits, world visits, and up-to-date weather maps (Griffin, 2010). Through these WWW mechanisms, a constructivist instructional model advance higher level instruction, such as problem solving and increased learner control. The WWW becomes a necessary tool for student-centered discovery and research. Of course, it can also be used for lower level drill and practice. At every level and in all subjects, students need to learn how to: precisely put questions, define contexts and purposes, pursue relevant information, analyze key concepts, derive sound inferences, generate good reasons, recognize questionable assumptions, trace important implications, and think empathically within different points of view (Dunn, 2010; hooks, 2010; Leicester, 2010). The WWW enables learners and teachers in each area by providing information for good reasoners to figure things out (Bowell; Levy, 2010). Critical thinking may be a key organizing concept for all educational reform (Bulach, Lunenburg, & Potter, 2012). Constructivism Constructivism is another, somewhat related, trend in education that can play a dynamic role in the relationship between how teachers teach and how children learn. One foundational premise of constructivism is that children actively construct their knowledge, rather than simply absorbing ideas spoken to them by teachers (Fosnot, 2006; Phillips, 2000; Larochelle, 2010). For example, Jean Piaget (1970) proposed that children make sense in ways very different from adults, and that they learn through the process of trying to make things happen, trying to manipulate their environment. Theories like these, which assert that “people are not recorders of information, but builders of knowledge structures,” have been grouped under the heading of constructivism (Pass, 2005; Wadsworth, 2004). Thus, students are ultimately responsible for NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 4____________________________________________________________________________________________ their own learning within a learning atmosphere in which teachers value student thinking, initiate lessons that foster cooperative learning, provide opportunities for students to be exposed to interdisciplinary curriculum, structure learning around primary concepts, and facilitate authentic assessment of student understanding. In constructivist theory, it is assumed that learners have to construct their own knowledge—individually and collectively. Each learner has a repertoire of conceptions and skills with which she or he must construct knowledge to solve problems presented by the environment. The role of the teacher and other learners is to provide the setting, pose the challenges, and offer the support that will encourage cognitive construction (Chaille, 2008). Since students lack the experience of experts in the field, teachers bear a great responsibility for guiding student activity, modeling behavior, and providing examples that will transform student group discussions into meaningful communication about subject matter (Flynn, 2005). Constructivism emphasizes the processes by which children create and develop their ideas. Applications lie in creating curricula that not only match but also challenge children’s understanding, fostering further growth and development of the mind (Baltes, 2007; Kincheloe, 2006; Leitner, 2010). Furthermore, when children collaborate in cooperative learning groups, they share the process of constructing their ideas with others. This collective effort provides the opportunity for children to reflect on and elaborate not only their own ideas but also those of their peers as well. With the improvement and access to the WWW, the children’s cooperative classroom becomes the world (Payne, 2010; Stewart, 2010). In this cooperative learning setting, children view their peers as resources rather than as competitors. A feeling of teamwork ensues. These processes have resulted in substantial advances in student learning (Bulach, Lunenburg, & Potter, 2012; Larochelle, 2010; Phillips, 2000). Constructivism is serving as the basis for many of the current reforms in several subject matter disciplines. The National Council of Teachers of Mathematics (NCTM) has published its document, Curriculum and Evaluation Standards for School Mathematics, which calls for mathematics classrooms where problem solving, concept development, and the construction of learner-generated solutions and algorithms are stressed rather than drill and practice on correct procedures and facts to get “the right” answer. The National Committee on Science Education Standards and Assessment similarly has issued its document, National Science Education Standards which calls for science education reform based on experimentation and learnergenerated inquiry, investigations, hypotheses, and models. The National Council of Teachers of English (NCTE) has called for emergent literacy as an important thrust in language arts reform. Interdisciplinary curricula is the theme of social studies reform being advocated by the National Council of Social Studies. Principles of Constructivist Pedagogy Jacqueline Brooks and Martin Brooks provide a detailed description of constructivist classroom practice and its theoretical underpinnings in their book, In Search for Understanding: The Case for Constructivist Classrooms (2005). They provide five principles of constructivist pedagogy: (a) posing problems of emerging relevance to learners; (b) structuring learning around “big ideas” or primary concepts; (c) seeking and valuing students’ points of view; (d) adapting curriculum to address students’ suppositions; and (e) assessing student learning in the context of teaching. FRED C. LUNENBURG ____________________________________________________________________________________________5 Principle 1: Posing problems of emerging relevance to students. Relevance does not have to be pre-existing for the student. Not all students come to the classroom interested in learning. Relevance can emerge through teacher mediation. Principle 2: Structuring learning around primary concepts. When designing curriculum, constructivist teachers organize information around conceptual clusters of problems, questions, and discrepant situations, because students are most engaged when problems and ideas are presented holistically rather than in separate, isolated parts. Much of traditional education breaks wholes into parts and then focuses separately on each part. But many students are unable to build concepts and skills from parts to wholes. Principle 3: Seeking and valuing students’ points of view. Students’ points of view are avenues into their reasoning. Awareness of students’ points of view help teachers challenge students, making school experiences both contextual and meaningful. Teachers who operate without awareness of their students’ points of view often doom students to dull, irrelevant experiences, and even failure. Principle 4: Adapting curriculum to address students’ suppositions. Teacher mediation is a key factor in adapting curriculum to address students’ suppositions. The teacher can abstract student learning or help build their own bridges from present understandings to new, more complex understandings. If suppositions are not explicitly addressed, most students will find lessons devoid of meaning, regardless of how charismatic the teacher or attractive the materials used. While it is the teacher who structures the opportunity, it is the students’ own reflective abstractions that create the new understanding. Principle 5: Assessing student learning in the context of teaching. Multiple-choice, norm-referenced tests are structured to determine whether students know information related to a particular body of knowledge. The overarching question posed by such activities is: What do you know?” Authentic assessment focuses on analytical thinking and performance, whereas norm-referenced, standardized tests focus on low-level rote skills. Becoming a Constructivist Teacher Brooks and Brooks (2005) provide the following set of descriptors of constructivist’ teaching behaviors, which they feel teachers can use to experiment with the approach. The set of descriptors describes teachers as facilitators of learning and empowerers of students to construct their own understandings of content, not simply as providers of information and managers of behavior. Constructivist teachers encourage and accept student autonomy and initiative. Autonomy and initiative cause students' pursuit of connections among concepts. Students who formulate questions and then go on to answer and analyze them are taking responsibility for their own learning and become problem solvers as well as problem finders. NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 6____________________________________________________________________________________________ Constructivist teachers use raw data and primary sources, along with manipulatives and interactive and physical materials. In the constructivist approach to teaching, learning becomes the result of research related to real problems. For example, students can be assigned to read historical accounts of the effects of social policies of the early 1980's on the economic profile of the African-American population in America. Or students can be taught to read the census reports and encouraged to generate their own inferences about social policies. The latter approach allows students to construct their own understandings of the issues. When framing tasks, constructivist teachers use cognitive terminology such as "classify", "analyze", "predict", and "create". Formulating tasks around cognitive activities such as analysis, interpretation, classification, and prediction, and explicitly using those terms with students, fosters the construction of new understandings about content. Constructivist teachers allow student responses to drive lessons, shift instructional strategies, and alter content. This does not mean that students' interest or lack of interest in a topic determines whether the topic is taught or that whole sections of the curriculum will be eliminated. It does mean that constructivist teachers will capitalize on "teachable moments" throughout the school year. These are moments when the students' interest, knowledge, and enthusiasm intersect and transcend a particular lesson. For example, the Persian Gulf War may have provoked student initiated discussion during that time period. Constructivist teachers inquire about students’ understandings of concepts before sharing their own understandings of those concepts. When teachers share their ideas before students have an opportunity to formulate their own, students' examination of their own ideas is eliminated. In such environments, most students will stop thinking about the concept and wait for the teacher to provide the "correct answer". Consequently, students are prevented from constructing their own ideas and theories. Constructivist teachers encourage students to engage in dialogue, both with the teacher and with one another. One way that students change or reinforce their ideas and theories is through social discourse. Students are empowered when they have an opportunity to present their own ideas and hear and reflect on the ideas of others. This process helps students construct new understandings or reflect on their existing ones. According to Robert Slavin (2009), student-to-student dialogue is the foundation upon which cooperative learning is based. Constructivist teachers encourage students’ inquiry by asking thoughtful, openended questions and encouraging students to ask questions of each other. Complex, thoughtful questions, that have more than one response, challenge students to delve into issues deeply and broadly and to form their own understandings of events and phenomena. Constructivist teachers seek elaboration of students’ initial response. Students' initial responses about issues are not necessarily their final thoughts, nor their best thoughts on a topic. Through elaboration of students’ initial responses, they frequently reconceptualize and assess their own errors and, in the process, construct their own understandings of issues, concepts, and theories. FRED C. LUNENBURG ____________________________________________________________________________________________7 Constructivist teachers engage students in experiences that might engender contradictions to their initial hypotheses and then encourage discussion. Cognitive growth occurs when an individual reformulates a current perspective. Students at all levels formulate and refine ideas about phenomena and then tenaciously hold onto these ideas as eternal truths. Even when confronted with authoritative evidence that challenge their views, students generally adhere to their original ideas. When teachers provide experiences that might engender contradictions, the framework for students' original ideas weaken, causing them to rethink their perspectives and formulate new understandings. Constructivist teachers allow wait time after posing questions. In most classrooms, there are some students who are not prepared to respond to questions or other stimuli immediately. They require more time to process information. Teachers that require immediate responses prevent these students from thinking through theories and concepts thoroughly, forcing them to become spectators. These students learn quickly that there is no point in mentally engaging in teacher-posed questions. Constructivist teachers provide time for students to construct relationships and create metaphors. Constructivist teachers structure and mediate classroom activities and provide the necessary time and materials for learning to occur, which causes students to construct patterns, relationships among concepts and theories for themselves. Constructivist teachers also encourage the use of metaphor as a way to facilitate learning. Metaphors help students to understand complex issues in a holistic way and to ruminate mentally with the parts of the whole to determine whether the metaphor works. Constructivist teachers nurture students' natural curiosity through frequent use of the learning cycle model. The learning cycle model has been used in science education for some time (Buxton, 2011). The model describes curriculum development and instruction as a threestep cycle: discovery, concept introduction, and concept application. First, the teacher provides an open-ended opportunity for students to interact with purposefully selected materials. This step is designed to generate student questions and hypotheses from working with the materials (discovery). Next, the teacher provides lessons aimed at focusing the students' questions, providing related and new vocabulary, framing with students their laboratory experiences, and such (concept introduction). Finally, students engage in one or more interactions of the discovery-concept introduction sequence. Students work on new problems with the potential for evoking a reflective, new look at the concepts studied previously (concept application). The aforementioned descriptors of constructivist teaching highlight practices that help students to construct their own understandings of challenging subject matter content. These descriptors can serve as guidelines for interpreting what it means to become a constructivist teacher. For specific examples of how to implement each of the descriptors, see Brooks and Brooks (2005). NATIONAL FORUM OF TEACHER EDUCATION JOURNAL 8____________________________________________________________________________________________ Conclusion NAEP data suggest that student outcomes in American education are a little better–and in some cases worse–than they were 30 years ago. Moreover, students in some other advanced, technological countries consistently outperform American students on international tests in science and mathematics. The ultimate goal of the No Child Left Behind legislation is that all students will demonstrate competency over challenging subject matter in the core subject areas— reading, mathematics, science, and social studies—and learn to use their minds well, so they are prepared for responsible citizenship, further learning, and productive employment in our Nation’s economy. Critical thinking and constructivism offer real promise for improving the achievement of all students in the core subject areas. References Baltes, P.B. (2007). Lifespan development and the brain: The perspective of biocultural and co-constructivism. West Nyack, NY: Cambridge University Press. Bowell, T. (2010). Critical thinking: A concise guide. New York, NY: Routledge. Brooks, J. G., & Brooks, M. (2005). In search of understanding: The case for constructivist classrooms. Alexandria, VA: Association for Supervision and Curriculum Development. Bulach, C.R., Lunenburg, F.C., & Potter, L. (2012). Creating a culture for highperforming schools: A comprehensive approach to school reform. Lanham, MD: Rowman & Littlefield. Buxton, C. A. (2011). Teaching science in elementary and middle school: A cognitive and cultural approach. Thousand Oaks, CA: Sage. Chaille, C. (2008). Constructivism across the curriculum: Big ideas as inspiration. Upper Saddle River, NJ: Allyn & Bacon. Cohen, E.D. (2010). Critical thinking. Lanham, MD: Rowman & Littlefield. Dunn, D.S. (2010). Teaching critical thinking: A handbook of best practices. New York: Wiley. Flynn, P. (2005). Applying standards-based constructivism: A two-step guide for motivating elementary students. Larchmont, NY: Eye on Education. Fosnot, C.T. (2006). Constructivism: Theory, perspectives, and practice. New York, NY: Teachers College Press. Griffin, R.E. (2010). Engaging creativity and critical thinking. Washington, DC: International Visual Literacy Association. Hooks, b. (2009). Teaching critical thinking. New York, NY: Routledge. Howard, T. G. (2011). Why race and culture matter in schools: Closing the achievement gap in America’s classrooms. New York: Teachers College Press. Kincheloe, J.L. (2006). Critical constructivism primer. New York, NY: Peter Lang. Larochelle, M. (2010). Constructivism and education. West Nyack, NY: Cambridge University Press. Leicester, M. (2010). Teaching critical thinking skills. London, England: Continuum International Publishing Group. FRED C. LUNENBURG ____________________________________________________________________________________________9 Leitner, L.M. (2010). Personal constructivism: Theory and applications. New York, NY: Pace University Press. Levy, D.A. (2010). Tools of critical thinking. Long Grove, IL: Waveland Press. Mason, M. (2010). Critical thinking and learning. New York, NY: Wiley. Moore, B. (2010). Critical thinking and formative assessments: Increasing the rigor in your classroom. Larchmont, NY: Eye on Education. Numrich, C. (2010). Raise the issues: An integrated approach to critical thinking. Upper Saddle River, NJ: Pearson. Paige, R. (2011). The black-white achievement gap: Why closing it is the greatest civil rights issue of our time. New York, NY: Amacom. Pass, Susan (2005). Parallel paths to constructivism: Jean Piaget and Lev Vygotsky. Charlotte, NC: Information Age. Payne, C.R. (2010). Information technology and constructivism in higher education: Progressive learning frameworks. Hersey, PA: IGI Global. Phillips, D.C. (2000). Constructivism in education, Vol. 1. Chicago, IL: University of Chicago Press. Piaget, J. (1970). Piaget’s theory. In P. Mussen (Ed.), Carmichael’s manual of child psychology (Vol. I, pp.703-732). New York: John Wiley. Slavin, R.E., & Madden, N.A. (2009). One million children: Success for all. Thousand Oaks, CA: Corwin Press. Stewart, C.M. (2010). Teaching and learning with technology: Beyond constructivism. New York, NY: Taylor & Francis. Tittle, P. (2010). Critical thinking: An appeal to reason. New York, NY: Taylor & Francis. U.S. Department of Education, National Center for Education Statistics. (2010a). NAEP assessments of fourth, eighth, and twelfth graders. Washington, DC: U.S. Government Printing Office. U.S. Department of Education. (2010b). The condition of education. Washington, DC: U. S. Government Printing Office. Vaughn, L. (2009). The power of critical thinking: Effective reasoning about ordinary and extraordinary claims. New York, NY: Oxford University Press. Wadsworth, B.J. (2004). Piaget’s theory of cognitive and affective development: Foundations of constructivism. Upper Saddle River, NJ: Allyn & Bacon. Copyright © 2009 by the National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in other formats without written permission from NCTM. Constructivist Learning and Teaching Douglas H. Clements and Michael T. Battista In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding. (MSEB and National Research Council 1989, 58) Radical changes have been advocated in recent reports on mathematics education, such as NCTM’s Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics 1989) and Everybody Counts (MSEB and National Research Council 1989). Unfortunately, many educators are focusing on alterations in content rather than the reports’ recommendations for fundamental changes in instructional practices. Many of these instructional changes can best be understood from a constructivist perspective. Although references to constructivist approaches are pervasive, practical descriptions of such approaches have not been readily accessible. Therefore, to promote dialogue about instructional change, each “Research into Practice” column this year* will illustrate how a constructivist approach to teaching might be taken for a specific topic in mathematics. What Is Constructivism? Most traditional mathematics instruction and curricula are based on the transmission, or absorption, view of teaching and learning. In this view, students passively “absorb” mathematical structures invented by others and recorded in texts or known by authoritative adults. Teaching consists of transmitting sets of established facts, skills, and concepts to students. Constructivism offers a sharp contrast to this view. Its basic tenets—which are embraced to a greater or lesser extent by different proponents—are the following: *1990–1991 6 1. Knowledge is actively created or invented by the child, not passively received from the environment. This idea can be illustrated by the Piagetian position that mathematical ideas are made by children, not found like a pebble or accepted from others like a gift (Sinclair, in Steffe and Cobb 1988). For example, the idea “four” cannot be directly detected by a child’s senses. It is a relation that the child superimposes on a set of objects. This relation is constructed by the child by reflecting on actions performed on numerous sets of objects, such as contrasting the counting of sets having four units with the counting of sets having three and five units. Although a teacher may have demonstrated and numerically labeled many sets of objects for the student, the mental entity “four” can be created only by the student’s thought. In other words, students do not “discover” the way the world works like Columbus found a new continent. Rather they invent new ways of thinking about the world. 2. Children create new mathematical knowledge by reflecting on their physical and mental actions. Ideas are constructed or made meaningful when children integrate them into their existing structures of knowledge. 3. No one true reality exists, only individual interpretations of the world. These interpretations are shaped by experience and social interactions. Thus, learning mathematics should be thought of as a process of adapting to and organizing one’s quantitative world, not discovering preexisting ideas imposed by others. (This tenet is perhaps the most controversial.) 4. Learning is a social process in which children grow into the intellectual life of those around them (Bruner 1986). Mathematical ideas and truths, both in use and in meaning, are cooperatively established by the members of a culture. Thus, the constructivist classroom is seen as a culture in which students are involved not only in discovery and invention but in a social discourse involving explanation, negotiation, sharing, and evaluation. Putting Research into Practice in the Elementary Grades: Readings from Journals of the NCTM 5. When a teacher demands that students use set mathematical methods, the sense-making activity of students is seriously curtailed. Students tend to mimic the methods by rote so that they can appear to achieve the teacher’s goals. Their beliefs about the nature of mathematics change from viewing mathematics as sense making to viewing it as learning set procedures that make little sense. Two Major Goals Although it has many different interpretations, taking a constructivist perspective appears to imply two major goals for mathematics instruction (Cobb 1988). First, students should develop mathematical structures that are more complex, abstract, and powerful than the ones they currently possess so that they are increasingly capable of solving a wide variety of meaningful problems. Second, students should become autonomous and self-motivated in their mathematical activity. Such students believe that mathematics is a way of thinking about problems. They believe that they do not “get” mathematical knowledge from their teacher so much as from their own explorations, thinking, and participation in discussions. They see their responsibility in the mathematics classroom not so much as completing assigned tasks but as making sense of, and communicating about, mathematics. Such independent students have the sense of themselves as controlling and creating mathematics. Teaching and Learning Constructivist instruction, on the one hand, gives preeminent value to the development of students’ personal mathematical ideas. Traditional instruction, on the other hand, values only established mathematical techniques and concepts. For example, even though many teachers consistently use concrete materials to introduce ideas, they use them only for an introduction; the goal is to get to the abstract, symbolic, established mathematics. Inadvertently, students’ intuitive thinking about what is meaningful to them is devalued. They come to feel that their intuitive ideas and methods are not related to real mathematics. In contrast, in constructivist instruction, students are encouraged to use their own methods for solving problems. They are not asked to adopt someone else’s thinking but encouraged to refine their own. Although the teacher presents tasks that promote the invention or adoption of more sophisticated techniques, all methods are valued and supported. Through interaction with mathematical tasks and other students, the student’s own intuitive mathematical thinking gradually becomes more abstract and powerful. Because the role of the constructivist teacher is to guide and support students’ invention of viable mathematical ideas rather than transmit “correct” adult ways of doing mathematics, some see the constructivist approach as inefficient, free-for-all discovery. In fact, even in its least directive form, the guidance of the teacher is the feature that distinguishes constructivism from unguided discovery. The constructivist teacher, by offering appropriate tasks and opportunities for dialogue, guides the focus of students’ attention, thus unobtrusively directing their learning (Bruner 1986). Constructivist teachers must be able to pose tasks that bring about appropriate conceptual reorganizations in students. This approach requires knowledge of both the normal developmental sequence in which students learn specific mathematical ideas and the current individual structures of students in the class. Such teachers must also be skilled in structuring the intellectual and social climate of the classroom so that students discuss, reflect on, and make sense of these tasks. An Invitation Each article in this year’s “Research into Practice” column will present specific examples of the constructivist approach in action. Each will describe how students think about particular mathematical ideas and how instructional environments can be structured to cause students to develop more powerful thinking about those ideas. We invite you to consider the approach and how it relates to your teaching—to try it in your classroom. Which tenets of constructivism might you accept? How might your teaching and classroom environment change if you accept that students must construct their own knowledge? Are the implications different for students of different ages? How do you deal with individual differences? Most important, what instructional methods are consistent with a constructivist view of learning? Constructivist Learning and Teaching 7 References Bruner, Jerome. Actual Minds, Possible Worlds. Cambridge, Mass.: Harvard University Press, 1986. Cobb, Paul. “The Tension between Theories of Learning and Instruction in Mathematics Education.” Educational Psychologist 23 (1988): 87–103. Mathematical Sciences Education Board (MSEB) and National Research Council. Everybody Counts: A 8 Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press, 1989. National Council of Teachers of Mathematics (NCTM). Curriculum and Evaluation Standards for School Mathematics. Reston, Va.: NCTM, 1989. Steffe, Leslie, and Paul Cobb. Construction of Arithmetical Meanings and Strategies. New York: Springer-Verlag, 1988. Putting Research into Practice in the Elementary Grades: Readings from Journals of the NCTM Constructivism and First-Grade Arithmetic Constance Kamii and Barbara A. Lewis For arithmetic instruction in the first grade, we advocate the use of games and situations in daily living in contrast to the traditional use of textbooks, workbooks, and worksheets. Our position is supported by the research and theory of Jean Piaget, called constructivism, as well as by classroom research (Kamii 1985, 1990). Piaget’s theory shows that children acquire number concepts by constructing them from the inside rather than by internalizing them from the outside. The best way to explain this statement is by describing children’s reactions to one of the tasks Piaget developed with Inhelder (Inhelder and Piaget 1963). The pupil is given one of two identical glasses, and the teacher takes the other one. After putting thirty to fifty chips (or beans, buttons, etc.) on the table, the teacher asks the pupil to drop a chip into his or her glass each time she drops one into hers. When about five chips have thus been dropped into each glass with one-to-one correspondence, the teacher says, “Let’s stop now, and you watch what I am going to do.” The teacher then drops one chip into her glass and says to the pupil, “Let’s get going again.” The teacher and the pupil drop about five more chips into each glass with one-to-one correspondence, until the teacher says, “Let’s stop.” The following is what has happened so far: Teacher: 1 !1 !1 !1 !1 !1 !1 !1 !1 !1 !1 Pupil: 1 !1 !1 !1 !1 !1 !1 !1 !1 !1 The teacher then asks, “Do we have the same amount, or do you have more, or do I have more?” Four-year-olds usually reply that the two glasses have the same amount. When we go on to ask, “How do you know that we have the same amount?” the pupils explain, “Because I can see that we both have the same amount.” (Some four-year-olds, however, reply that they have more, and when asked how they know that they have more, their usual answer is “Because.”) The teacher goes on to ask, “Do you remember how we dropped the chips?” and four-year-olds usually give all the empirical facts correctly, including the fact that only the teacher put an additional chip into her glass at one point. In other words, four-yearolds remember all the empirical facts correctly and base their judgment of equality on the empirical appearance of the two quantities. By age five or six, however, most middle-class pupils deduce logically that the teacher has one more. When we ask these pupils how they know that the teacher has one more, they invoke exactly the same empirical facts as the four-year-olds. No one teaches five- and six-year-olds to give correct answers to these questions. Yet children all over the world become able to give correct answers by constructing numerical relationships through their own natural ability to think. This construction from within can best be explained by reviewing the distinction Piaget made among three kinds of knowledge according to their sources—physical knowledge, logicomathematical knowledge, and social (conventional) knowledge. Physical knowledge, on the one hand, is knowledge of objects in external reality. The color and weight of a chip are examples of physical properties that are in objects in external reality and can be known empirically by observation. Logicomathematical knowledge, on the other hand, consists of relationships created by each individual. For instance, when we are presented with a red chip and a blue one and think that they are different, this difference is an example of logicomathematical knowledge. The chips are observable, but the difference between them is not. The difference exists neither in the red chip nor in the blue one, and if a person did not put the objects into this relationship, the difference would not exist for him or her. Other examples of relationships the individual can create between the chips are similar, the same in weight, and two. Physical knowledge is thus empirical in nature because it has its source partly in objects. Logicomathematical knowledge, however, is not Constructivism and First-Grade Arithmetic 9 empirical knowledge, as its source is in each individual’s head. The ultimate sources of social knowledge are conventions worked out by people. Examples of social knowledge are the fact that Christmas comes on 25 December and that a tree is called “tree.” Words such as one, two, and three and numerals such as 1, 2, and 3 belong to social knowledge, but the numerical concepts necessary to understand these numerals belong to logicomathematical knowledge. Keeping the distinction among the three kinds of knowledge in mind, one can understand why most four-year-olds in the task described earlier said that the two glasses have the same amount. The four-yearolds had not yet constructed the logicomathematical relationship of number and could therefore gain only physical knowledge from the experience. From the appearance of the chips in the glasses, the pupils concluded that the amount was the same despite the fact that they remembered the way in which the chips had been dropped. Once the concept of number has developed, however, pupils will deduce from the same empirical facts that the teacher has one more chip regardless of the physical appearance. New Principles of Teaching The following principles of teaching flow from constructivism and the preceding goals: 1. Encourage pupils to invent their own ways of adding and subtracting numbers rather than tell them how. For example, if pupils can play a board game with one die, we simply introduce a second die and let them figure out what to do. 2. Encourage pupils to exchange points of view rather than reinforce correct answers and correct wrong ones. For example, if a pupil says that six minus two equals three, we encourage pupils to agree or disagree with each other. Pupils will eventually agree on the truth if they debate long enough because, in logicomathematical knowledge, nothing is arbitrary. 3. Encourage pupils to think rather than to compute with paper and pencil. Written computation interferes with pupils’ freedom to think and to remember sums and differences. Classroom Activities New Goals for Beginning Arithmetic Instruction If children develop mathematical understanding through their own natural ability to think, the goals of beginning arithmetic must be that pupils think and construct a network of numerical relationships. To add five and four, for example, pupils have to think (1 ! 1 ! 1 ! 1 ! 1) ! (1 ! 1 ! 1 ! 1). This operation requires pupils to make two wholes (5 and 4) in their heads and then to make a higher-order whole (9) in which the original wholes (5 and 4) become parts. An example of a network of numerical relationships can be seen when pupils think about 5 ! 4 as one more than 4 ! 4 and as one less than 5 ! 5. Addition thus involves a great deal of thinking, that is, the making of relationships rather than mere skills (such as penmanship). This definition of goals for instruction is very different from traditional instruction that focuses on correct answers and the writing of mathematical symbols. It is also very different from the assumption that pupils have to internalize “addition facts,” store them, and retrieve them in computerlike fashion. 10 Paper-and-pencil exercises cause social isolation, mechanical repetition, and dependence on the teacher to know if an answer is correct. We, therefore, replace the textbook, workbook, and worksheets with two kinds of activities: games and situations in daily living. Games, such as a modification of old maid in which pupils try to make a sum of ten with two cards, are well known to be effective. Although games are typically used only as a reward for pupils who have finished their work, we use games as a staple of instruction. Games give rise to compelling reasons for pupils to think and to agree or disagree with each other. When it is useful to know that 5 ! 5, 6 ! 4, 7 ! 3, and so on, all equal ten, pupils are much more likely to remember these combinations than when they write in workbooks to satisfy the teacher. Situations in daily living also offer meaningful opportunities for pupils to construct mathematical relationships. Taking attendance, voting, collecting money, and sending notes home are examples of situations the teacher can use to encourage pupils to think. If four people brought their lunch, eight Putting Research into Practice in the Elementary Grades: Readings from Journals of the NCTM ordered the special, and six ordered soup and sandwich, the teacher can ask if everybody present has been accounted for. Pupils care about real-life situations and think much harder about these questions than about those in workbooks. References Inhelder, Bärbel, and Jean Piaget, “De I’itération des actions à la récurrence Elémentaire.” In La formation des raisonnements récurrentiels, edited by Pierre Greco, Bärbel Inhelder, B. Matalon, and Jean Piaget. Paris: Presses Universitaires de France, 1963. Kamii, Constance. “Constructivism and Beginning Arithmetic (K–2).” In Teaching and Learning Mathematics in the 1990s, 1990 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Thomas J. Cooney and Christian R. Hirsch, pp. 22–30. Reston, Va.: NCTM, 1990. ——. Young Children Reinvent Arithmetic. New York: Teachers College Press, 1985. Constructivism and First-Grade Arithmetic 11
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