A Cognitive/Constructivist Approach in Math

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1. For me a cognitive/constructivist approach to learning about Math is based on giving students the support and the opportunity to construct their own meaning about what they are learning, or constructing their own meaning from the experience of learning. In math, the teacher's role is to provide the environment and the experiences that will offer that opportunity to the student, and the support to fully untilize the experience to learn. These experiences can best be provided through lessons where the environment and the experiences are all set up and designed with opportunities for the learner, or learner-centered instruction. Piaget stated that the environment is passive and that it is the learner's interactions with that environment that allows that learneer to construct his or her own meaning from that experience. Cooperative learning also allows students to communicate with peers in order to construct their own meaning from not just the environment, but alos from an interaction or experience with others trying to extract their own meaning about the same concepts.

2. Cooperative Learning, Collaborative Learning, Learning Centers, Problem Solving Approach to learning, Learning Centered Instruction, Using Manipulatives, Using the classroom walls, posters, charts, games, and art, in an ineractive experiential manner (filling the room with knwledge and information the students can use and interact with), and a facilitatitive approach to teaching math including dialogue and consensus about math.

3. A cognitive/constructivist approach consists of interactions between students/students and teacher/students (Van Zoest et al., 1994), mathematical dialogue and consensus between students (Van Zoest et al., 1994), teachers providing just enough information to establish background/intent of the problem, and students clarifing, interpreting, and attempting to construct one or more solution processes (Cobb et al., 1991),
teachers accepting right/wrong answers in a non-evaluative way (Cobb et al., 1991), teachers guiding, coaching, asking insightful questions and sharing in the process of solving problems (Lester et al., 1994),
teachers knowing when it is appropriate to intervene, and when to step back and let the pupils make their own way (Lester et al., 1994). A further characteristic is that a problem-solving approach can be used to encourage students to make generalizations about rules and concepts, a process which is central to mathematics (Evan and Lappin, 1994).
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LINK: http://pages.uoregon.edu/moursund/Math/learning-theories.htm#Constructivism

Constructivism and Learning Mathematics

Howard Gardner has identified Logical/mathematical as one of the eight (or more) intelligences that people have. As with the other intelligences in Gardner's classification system, people vary considerably in the innate levels of mathematical intelligence that they are born with.

People like to argue nature versus nurture in terms of both general intelligence and intelligence within specific domains such as those that Gardner lists. We know that the brain has great plasticity, that there is a lot of brain growth after a person is born, that the brain continues to grow new neurons and new connections among neurons throughout life, that certain drugs can damage brain cells, that proper nutrition is needed for proper brain growth, and so on.

A certain amount of math knowledge and skill is innate--genetic in origin. The great majority of a person's math knowledge and skills comes from learning--learning to use parts of the brain that can learn to do math, but were not genetically designed specifically for this purpose.

Math is a cumulative, vertically structured discipline. One learns math by building on the math that one has previously learned. That, of course, sounds like Constructivism.

In brief summary, here is a constructivist approach to thinking about mathematics education.

People are born with an innate ability to deal with small integers (such as 1, 2, 3, 4) and to make comparative estimates of larger numbers (the herd of buffalo that we saw this morning is much smaller than the herd that we are looking at now.)
The human brain has components that can adapt to learning and using mathematics.
Humans vary considerably in their innate mathematical abilities or intelligence.
The mathematical environments that children grow up in vary tremendously.
Thus, when we combine nature and nature, by the time children enter kindergarten, they have tremendously varying levels of mathematical knowledge, skills, and interests.
Even though we offer a somewhat standardized curriculum to young students, that actual curriculum, instruction, assessment, engagement of intrinsic and extrinsic motivation, and so on varies considerably.
Thus, the are huge differences among the mathematical knowledge and skill levels of students at any particular grade level or in any particular math course. In addition, there are considerable differences in their ability to learn mathematics.
Thus, mathematics curriculum, instruction, and assessment needs to appropriately take into consideration these differences. One way to do this is through appropriate use of constructivist teaching and learning principles.
It is interesting to note that many researchers and practitioners in ICT have come to the same conclusion about teaching and learning ICT. They recommend a constructivist approach.

Journaling, Project-based Learning, and Problem-based Learning are all standard components of a constructivist teaching/learning environment. (Note that both project-based and problem-based learning are abbreviated PBL.) One of the strands of this workshop is devoted to ICT-Assisted PBL

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Classroom Applications of Constructivism

LINK: http://www.teach-nology.com/currenttrends/constructivism/classroom_applications/

Learning theory of constructivism incorporates a learning process wherein the student gains their own conclusions through the creative aid of the teacher as a facilitator. The best way to plan teacher worksheets, lesson plans, and study skills for the students, is to create a curriculum which allows each student to solve problems while the teacher monitors and flexibly guides the students to the correct answer, while encouraging critical thinking.

Instead of having the students relying on someone else's information and accepting it as truth, the students should be exposed to data, primary sources, and the ability to interact with other students so that they can learn from the incorporation of their experiences. The classroom experience should be an invitation for a myriad of different backgrounds and the learning experience which allows the different backgrounds to come together and observe and analyze information and ideas.

Hands-on activities are the best for the classroom applications of constructivism, critical thinking and learning. Having observations take place with a daily journal helps the students to better understand how their own experiences contribute to the formation of their theories and observational notes, and then comparing them to another students' reiterates that different backgrounds and cultures create different outlooks, while neither is wrong, both should be respected.

Some strategies for classroom applications of constructivism for the teacher include having students working together and aiding to answer one another's questions. Another strategy includes designating one student as the "expert" on a subject and having them teach the class. Finally, allowing students to work in groups or pairs and research controversial topics which they must then present to the class.
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Douglas H. Clements
State University of New York at Buffalo, Buffalo, NY 14260

LINK: http://investigations.terc.edu/library/bookpapers/constructivist_learning.cfm

Michael T. Battista
Kent State University, Kent, OH 44242

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 Mathematics1989) 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 ...