The
traditional mathematics and classes have been taught with the predictable
lecture and drill approach. Students are given specific methods to solve
problems and then practice each technique on a large number of problems.
The emphasis is placed on getting the one correct answer and following
the teacher's steps in the solution process. Even though this is a common
teaching method, there is much evidence to support its ineffectiveness
in promoting long-term learning and solid understanding of concepts.
Constructivism
Constructivism is the belief that students build their own knowledge by integrating new information and ideas into existing knowledge structures. When there are inconsistencies between the new and old ideas, the learner analyzes what he/she believes to be true against new ideas. In the process, some old ideas may be replaced or restructured with new notions or new ideas discarded as invalid.
Learning occurs when students build their own understandings by making connections among new ideas and existing knowledge.
Constructivism has several implications for teaching. First, learning does not occur unless students perceive discrepancies. They must have the opportunity to explore tasks that provide tension or dissonance with their "old" knowledge.
Second, the idea that all students leave the classroom with the same information if the teacher gives clear and succinct explanations is not valid. No matter the detail, students put new ideas into their already existing knowledge. They may not perceive what teachers say in the same way that teachers meant it to be.
Third, students need to share ideas in order to validate or refute their understandings or beliefs. They should have opportunities to discuss their own ideas and analyze others. The instructional approach changes from one of "teacher as authority" to "teacher as facilitator." Additionally, mathematical tasks should be open-ended with multiple solutions and solution strategies rather than close-ended problems with only one correct answer and a step-by-step solution method. These open-ended tasks motivate discussion and encourage exploration.
Finally, teachers must acknowledge that students structure and integrate their knowledge through a variety of modalities, such as linguistic, spatial, mathematical and musical. That is, students have different strengths and need to have a variety of teaching strategies used in the class in order to find a modality or method that fits how they structure their knowledge.
Constructivism supports five specific premises (Vygotsky, 1978) that optimize learning experiences. These include communication, problem solving, connections within and outside mathematics, time for learning, and challenges for all students.
Communication (social interaction), problem solving, connections within and outside of mathematics and science, time for learning, and open-ended challenges optimize learning experiences for all students.
Communication. Social interaction is the means by which we convey our ideas and make new conjectures. For many of us, social interaction conjures up visions of students discussing after-school activities, but it has a different connotation in a classroom that emphasizes student learning. Social interaction provides the means for students to use their language to communicate original ideas about mathematics and science.
Social interaction appears in at least five forms. First, the most obvious form of social interaction is oral discussion. Students describe, verify, and challenge solution methods and results of investigations. The teacher acts as the facilitator, asking questions that probe, clarify, or challenge student ideas.
Second, social interaction is accomplished through writing tasks. Students can be given journal prompts and their responses are indicative of their level of mathematical and scientific development. Journal prompts can relate to the content, to student feelings about themselves as mathematicians and scientists and students of mathematics and science, to student views about mathematics and science as a discipline, and to metacognitive aspects about solution processes students use. Additionally, students can create new problems for other students to solve or they can write detailed descriptions about how they solved a nonroutine problem.
Third, social interaction occurs through listening. As students share their ideas, other students must develop listening skills, so that they can analyze what others are saying. Their analysis forms the basis for refuting or verifying others' ideas.
Reading is the fourth form of social interaction. Mathematics and science classes can use resources other than the textbook. These may include trade books, reference books, or student-created problems. Students can use the resources as support for their ideas, much as one would in the workplace.
Finally, building models is another form of communication. Models can be constructed from commercial materials or from any items that students feel communicate important characteristics of a mathematical and scientific topic.
Problem solving. In the past, students learned skills first and, only after they had mastered algorithms, were they given the opportunity to engage in problem-solving tasks. The learning of skills through routinized methods did not promote student creativity nor did it help students learn to attack nonroutine or unfamiliar problems. These standards provide a different approach to problem solving. That is, they recommend that students first be exposed to problem solving and, from these experiences, move to skills.
Problem-solving tasks allow students to bridge what they have experienced into new areas. Good problems give students the opportunity to use whatever they feel comfortable with in the solution process. Thus, students may choose a variety of problem-solving strategies that best fit their way of learning.
Additionally, students develop a growing knowledge about how they are learning. Many adults have never been "in touch" with their thinking as they solve problems. They may not realize that they gravitate to certain solution processes. This knowledge helps students become better problem solvers.
Connections. Closely associated with problem solving is connections, a linking between old and new knowledge. As students confront new problem contexts, they look to their old experiences to help define ways to solve the new problems. The fluidity of problem contexts develops the strong connections.
It is important for students to form connections among their knowledge bases. Mathematics and science has been typically taught as a set of fragmented and isolated skills that students forget when a new skill is introduced. The connections among topics help students determine the rationale behind particular algorithms and, hence, solidify the learning of otherwise meaningless mathematics.
Time for learning. For decades, mathematics and science teachers have drilled students in computational methods. The primary method of learning was to practice, practice, and then do more practice. If, however, a teacher were working with a "faster" student, one who scored high on a standardized test, then perhaps the pace was faster.
We now know that regardless of ability level, students need approximately three to eight days to really understand a new idea. That does not mean that it is the only topic that students study during that time but it does imply that students should see problems related to a topic over a period of days and not for just one night. This supports the call for more in-depth work in concepts rather than a superficial treatment of a topic.
Challenge for all students. One of the goals of the standards is to allow all students the opportunity to succeed in mathematics and science. Veteran and inexperienced teachers often claim that it is not possible to challenge all students in the same class. One method that does encourage students to work at their own level is the use of open-ended questions.
Open-ended questions allow students to rise to the height of their understandings in order to respond. Multiple-solution responses encourage students to continue to find more complex answers. And, because there are multiple answers, students persevere on the problems longer.
The result is that students become more confident of their mathematical and scientific abilities and they are more willing to share their responses. Challenged students are more engaged in class discussions and look forward to accepting new and bigger challenges.