Constructivsm in Mathematics Education


Constructivism in mathematics education? The most influential and widely accepted philosophical perspective in mathematics education today is constructivism. This view, which holds that individuals construct their own knowledge, can be traced back to Piaget and beyond. While it takes many forms, at its simplest, it sees the learner as an active participant, not as a blank slate upon which we write or as an empty vessel which we fill. In this view, cognition is considered adaptive, in the sense that it tends to organize experiences so they “fit” with a person’s previously constructed knowledge.

As a consequence, both researchers and teachers ask, “What is going on in students’ minds when . . . ?”, rather than speaking of behavioral outcomes and asking, “Which stimulus will elicit a desired response?” The term “constructivism” often designates this view of how people learn, and constructivist teaching often simply means taking students’ views and background into account so as to engender active, meaningful learning. However, constructivism comes in a variety of “flavors.” There is a “moderate” version, compatible with the way most mathematicians see mathematics, and a social constructivist version, inspired by the work of Vygotsky, which takes into account sociocultural perspectives. There is the radical constructivism of von Glasersfeld, and beyond that, the sociology of scientific knowledge (SSK), which replaces the idea of truth with that of utility. We will describe these views and place them along a (increasingly radical) continuum from the above “plain vanilla” version, which almost everyone in mathematics education today accepts and tries to act on, through the radical view, to the very relativistic SSK. Adopting the “plain vanilla” view that students construct (or reconstruct) knowledge for themselves does not prescribe a single “constructivist way of teaching.” It does, however, suggest that lecturing is likely to be less effective, than more active approaches such as cooperative group learning, and as a result, students are likely to make weak constructions. Ways of teaching that require students, not only to be more active, but to reflect on their work, are likely to encourage them to make strong constructions that result in increased conceptual knowledge and more connections. We will indicate some general principles this constructivist view entails. John Selden, MERC, Box 2781, Cookeville, TN 38502 Annie Selden, Tennessee Technological University, Cookeville, TN 38505 Constructivism “Constructivism” is a philosophical viewpoint on how the mind forms and modifies its understanding of reality. It is the foundation of our outlook on pedagogy and research. In what way is a constructivist view of science education different from other views? The answer lies in the tenets of constructivist philosophy, which assert that all knowledge is constructed as a result of cognitive processes within the human mind. While this may appear to be a harmless enough statement, many find (so-called) radical constructivism somewhat unpalatable. Radical constructivism challenges the notion of an external reality: No amount of stimuli, experience, or thinking is sufficient to prove the existence of an external agent. Science (of course) presumes such an external reality and seeks to describe its nature and behavior. (Science also presumes that the external reality is well behaved and capable of being explained.) Despite the previous statements, there is no essential conflict between science and constructivism at the operational level. In fact, scientists readily admit that all we can ever do is construct a model of external reality, assuming it exists. Thus, all that we know is actually a set of stimuli and experiences. This is totally in accord with the scientific view. So, at the level of epistemology (how we know or learn anything), science and constructivism are in complete harmony. The premises of constructivism as an epistemology are: 1. Knowledge is constructed, not transmitted. 2. Prior knowledge impacts the learning process. 3. Initial understanding is local, not global. 4. Building useful knowledge structures requires effortful and purposeful activity. The constructivist perspective is clearly divergent from earlier views of education that presumed we could put or pour information directly into a student’s head. Starting from constructivism, real learning can occur only when the learner is actively engaged in operating on, or mentally processing, incoming stimuli. Furthermore, the interpretation of stimuli depends upon previously constructed learning. Nothing here should be taken to imply that the mental processing involved in learning is necessarily conscious. In fact, much, perhaps even most, of the learning we do is subconscious. Thinking or learning about the process of learning, rather than the material being learned, is often called a meta-cognitive process. Cognitive science has undertaken the study of the mental processes used to acquire, store, process, and use knowledge. Essential to any such study is a theory of learning and cognition. As a theory of epistemology, constructivism plays a central role in cognitive science, a role akin to that of causality for the physical sciences. Like causality, constructivism provides no specific answers, but rather, frames the questions and the acceptable forms of answers. In addition to being used as a philosophy and an epistemology, constructivism also can be used to indicate a theory of communication. When you send a message by saying something or providing information, if you have no knowledge of the receiver, then you have no idea as to what message was received, and you can not unambiguously interpret the response. Viewed in this way, instruction becomes the establishment and maintenance of a language and a means of communication between the teacher and students, as well as between students. Simply presenting material, giving students problems, and accepting answers back is not a refined enough process of communication for efficient learning. For pedagogic purposes, the tenets of constructivism can be rephrased as follows: 1. Students come into our classrooms with an established world-view, formed by years of prior experience and learning. 2. Even as it evolves, a student’s world-view filters all experiences and affects their interpretation of observations. 3. Students are emotionally attached to their world-views and will not give up their world-views easily. 4. Challenging, revising, and restructuring one’s world-view requires much effort. If we base instruction on the principles of constructivism, the role of the teacher is raised from someone who simply dispenses information to someone who structures activities that improve communication, that challenge students’ pre-conceived notions, and that help students revise their world-views. In spite of the difficulties, cognitive research has been able to identify important patterns in the ways students and experts think about their subjects, suggesting pedagogic practices that enhance learning. Collaborative Group Techniques A discussion of teaching via small-group cooperative learning work. William J. Leonard Robert J. Dufresne William J. Gerace Jose P. Mestre January 1999 Supplement A of the “Teacher’s Guide to accompany Minds*On Physics: Motion”. ________________________________________ Introduction Collaborative groups and cooperative learning refer to a variety of structured classroom management techniques and grading systems developed and studied by Aronson, Johnson & Johnson, Kagan, Slavin, and others since the early 1970s. These terms usually do not refer to loosely structured group work in which students are told simply to “work together” on a problem or assignment. To emphasize the difference between unstructured group work and collaborative group work, groups are usually referred to as teams. Collaborative structures are content-free, and thus can be used in a variety of contexts. Studies have shown that in well structured cooperative groups, students consistently learn many different subjects better than students in traditionally structured classrooms. Cooperative learning also has a number of psychological and social benefits, such as being exposed to other points-of-view, learning how to cooperate, having more positive feelings about school, having more positive feelings about themselves and others, and wanting their classmates to do well. Studies have shown also that all students benefit academically from cooperative learning. Successful students show modest gains in performance and achievement, while historically unsuccessful students usually show tremendous gains when taught using cooperation as the primary motivator. Cooperative grouping lets students organize their thoughts in a less threatening context than whole-class discussions, and prepares students for sharing their thoughts with the class. Also relevant for Minds•On Physics is that students can make progress on exercises they would not be able to attempt alone. Getting started with cooperative learning Because students often lack collaborative group skills, it is essential to begin the school year with activities designed to target interaction skills and team building within the class. Students need to learn how to listen to other students, and to analyze and interpret what they are saying. Students must learn, for example, how to encourage others in their group to participate, how to ask questions, how to manage dominant personalities, how to monitor and modify the group dynamic, and how to communicate effectively. Unless these skills are targeted early in the year, cooperative learning is likely to fail. Therefore, the focus of instruction at the beginning of the year should be on developing group skills, rather than on physics. This investment of time will yield huge dividends later in the year. For example, have students sit in a circle and have volunteers define what they think science is all about. Then require the person sitting on his/her left (or right, whichever you choose) to paraphrase the definition. Be sure to tell students the structure of the activity beforehand, and have the class discuss and reflect on the activity immediately afterwards. Another effective structure is to have a team of three or four students work on a problem together — a problem from algebra, for example, that they should already know how to solve — and have three or four other students observe the interactions. Afterwards, have everyone discuss what happened, and what didn’t happen, as the inner group solved the given problem. (This is sometimes called Fishbowl.) Some common collaborative group structures There are literally hundreds of cooperative structures and dozens of books available to help teachers incorporate cooperative learning into their classrooms. The structures listed and described here are believed to be particularly useful with the Minds•On Physics materials and approach. Fishbowl. Teams of three or four work on a problem or exercise. At the same time, other teams of three or four observe the first teams. In particular, the first teams work on seeking other points-of-view, listening to and paraphrasing ideas, and other communication skills while solving the given problem. The second teams focus their attention on the team dynamic and make sure they are prepared to discuss how well or poorly the first teams worked together to solve the problem. (There is sometimes the tendency of the second teams to focus on the problem rather than the team dynamic.) After some time (even if every team has not finished the problem), the class discusses what happened and what didn’t happen during the activity. Pairs Check. Teams of four work in pairs on a set of exercises. First one member works on a problem, while the second member coaches. Then the second member works on a problem while the first coaches. Pairs then check their answers with members of the same team. After all problems, inconsistencies, etc. are resolved, the process is repeated for subsequent exercises. Pairs Check II. This is the same as Pairs Check, except that students do exercises individually beforehand. One student explains his/her answer on a question to another student, and they discuss it. Then, they reverse roles for the next question. Answers are agreed upon before sharing with the whole team. Teams Check. Teammates help each other understand answers to exercises, so that any member of the team may be called upon to answer any one of the questions. Jigsaw. If there is reading material (such as background) to be digested before doing an activity, split it up into 3 or 4 self-contained parts. Divide the class into the same number of Reading Groups, with one member from each team. Give one part of the reading to each team to digest and to prepare to explain to their team. Then rearrange the students so that each team has someone who has read one of the self-contained parts, and have each student teach his/her part of the reading to the rest of the team. Think-Pair-Share. Students think about each question, pair off and discuss the question with a classmate, and share their answers with the class. Think-Pair-Square. This is the same as Think-Pair-Share, except that students share their answers with members of another pair. Word Webbing. As a team or individually, open-ended or with concepts provided by the teacher, students construct a concept map within a specified domain. If done in teams, each member should have a different color of pen. Team Product. Students work together, but each has a primary role within the team. Some favorite roles are: Manager (to keep the team on task); Reader (to read aloud the question being answered by the team); Encourager (to make sure everyone participates); Checker (to make sure everyone understands); Writer (to record results and to make sure everyone agrees); Artist (if needed to prepare the presentation); and Presenter (if needed to explain the team’s answer to the rest of the class). The most important roles are the Manager, Checker, and Writer. (In other words, teams of three with these three roles are the most common.) In order to accommodate some of the other roles, students may take on two of them at the same time. Blackboard Share. Teams share their answers with the class and get feedback. This can be done on posterboards or transparencies instead. Roving Reporter. When a team gets stuck, one member is allowed to roam the room looking for ideas and reports back to the team. Two-Box (or Two-Column) Induction. The teacher puts items into one of two boxes (usually on the blackboard) without telling students what the criteria are for sorting the items. As the teacher adds items to the two boxes, students (standing in teams) discuss the items and possible categories. When a team decides that they know how the sort was done, they sit down without revealing their answer. (This is a non-disruptive way of letting the teacher know how the individual teams are doing.) When all teams are seated, there are three different options: 1. Ask each team to add an item to each box, and let the other teams evaluate and comment on the choices; 2. Present additional items to the class, and ask teams to decide which box each item belongs in. 3. Have teams describe their categories. Deciding which structure should be used Some structures are more compatible with certain activities or instructional goals than others. For instance, Fishbowl is good for developing skills; Pairs Check and Jigsaw are good for learning new material; and Word Webbing and Two-Box Induction are good for relating concepts. Also, do not introduce too many new structures too quickly; usually about one new structure per week is recommended. Other advice It is usually a good idea to have the details of a cooperative group activity worked out before class. You should know how students are to divide themselves into teams (e.g., by assignment, by drawing lots, or by personal preference); how many students should be on each team (2, 3, or 4, usually); what the team composition should be (heterogeneous or homogeneous; all male, all female, or mixed); which questions each team should work on; when the activity is officially over; how to bring the activity to closure; and how to grade the activity (if at all). Also, it’s important that everyone has an active role within each team, and that there are “sponge” activities that teams can work on if they finish earlier than other teams. Finally, hang in there. It takes some perseverance for both students and teachers to get collaborative groups to work effectively, but the rewards are definitely worth the effort. References The following books and articles should help teachers incorporate collaborative learning. In particular, Cooperative Learning by Spencer Kagan has an entire chapter devoted to resources, including books on theory, research, and methods, manipulatives, video tapes, newsletters, and the names and addresses of some cooperative learning organizations. REFFERNCES Aronson, E., Blaney, N., Stephan, C., Sikes, J. & Snapp, M. (1978). The Jigsaw Classroom. Beverly Hills, CA: Sage Publications. Johnson, R.T. & Johnson, D.W. (1991). Learning Together and Alone: Cooperative, Competitive, and Individualistic Learning (3rd ed.). Boston, MA: Allyn & Bacon. Johnson, R.T., Johnson, D.W. & Holubec, E.J., Eds. (1987). Structuring Cooperative Learning: Lesson Plans for Teachers. Edina, MN: Interaction Book Company. Johnson, R.T., Johnson, D.W. & Holubec, E.J. (1993). Circles of Learning: Cooperation in the Classroom. Edina, MN: Interaction Book Company Johnson, R.T., Johnson, D.W. & Smith, K.A. (1991). Active Learning: Cooperation in the College Classroom. Edina, MN: Interaction Book Company. Heller, P., Keith, R. & Anderson, S. (1992). Teaching problem solving through cooperative grouping. Part 1: Group versus individual problem solving. American Journal of Physics, 60, 627-636. Heller, P. & Hollabaugh, M. (1992). Teaching problem solving through cooperative grouping. Part 2: Designing problems and structuring groups. American Journal of Physics, 60, 637-644. Kagan, S. (1990). The structural approach to cooperative learning. Educational Leadership, 47(4): 12-15. —— (1996). Cooperative Learning. San Clemente, CA: Kagan Cooperative Learning. Sharan, S., Hare, P., Webb, C.D. & Hertz-Lazarowitz, S., Eds. (1980). Cooperation in Education. Provo, UT: Brigham Young University Press.

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