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How does conceptual knowledge relate to procedural knowledge? And, can teachers mediate conceptual development? Science and mathematics educators have become increasingly aware that our understanding of conceptual change is at least as important as the analysis of the concepts themselves.

In fact, a plethora of research has established that concepts are mental structures of intellectual relationships, not simply a subject matter. The research indicates that the mental structures of intellectual relationships that make up mental concepts organize human experiences and human memory Bartsch, Therefore, conceptual changes represent structural cognitive changes , not simply additive changes. Based on the research in cognitive psychology, the attention of research in education has been shifting from the content e.

Despite the research, many teachers continue to approach new concepts as if they were simply addons to their students' existing knowledge—a subject of memorization and recall. This practice may well be one of the causes of misconceptions in mathematics. The notion of structural cognitive change, or schematic change , was first introduced in the field of psychology by Bartlett, who studied memory in the s. It became one of the basic tenets of constructivism. Researchers in mathematics education picked up on this term and have been leaning heavily on it since the s, following Skemp , Minsky , and Davis The generally accepted idea among researchers in the field, as stated by Skemp , p.

It involves the whole network of interrelated operational and conceptual schemata. Structural changes are pervasive, central , and permanent.

Conceptual and procedural knowledge : the case of mathematics

The first characteristic of structural change refers to its pervasive nature. That is, new experiences do not have a limited effect, but cause the entire cognitive structure to rearrange itself. Vygotsky , p. The development of each function, in turn, depends upon the progress in the development of the interfunctional system. Neuroscientists describe the pervasiveness of change by referring to the neuroplasticity of the brain.

A new experience causes new connections to form among the dendrites and axons attached to the brain's cells and changes the structure of the brain. When a cognitive change is structural, the structure as a whole is affected.

Conceptual v.s Procedural Knowledge

Mathematical thinking is viewed as a structure of a connected collection of hierarchical relations. The second characteristic of structural cognitive change is centrality , or the autonomous, self-regulating propensity of the change. Simply stated, when one learns something and that learning results in structural change, one is prepared to learn something more advanced in the same category.

As this example shows, new structures act to secure themselves as they accommodate new experiences. This characteristic of structural cognitive change best explains the open-ended and continuous development of a person's cognition.

The Case of Mathematics, 1st Edition

Is that the case? Mathematicians are the first to counter this argument by stressing that much of mathematics is based on deductive proof, not on exploration and experimentation. Most of us would assert that our own mathematics knowledge has indeed been acquired not through our own private research, but under the at times forceful guidance of mathematically educated people around us. The fact that the majority of people develop a mathematical knowledge that represents almost the entire history of this discipline proves that the question is moot.

Rather than approaching the problem as an input-output dichotomy of choices, it is important to examine how more constructive the solution becomes when it is focused on the process of learning. In mathematics, conceptual knowledge otherwise referred to in the literature as declarative knowledge involves understanding concepts and recognizing their applications in various situations.

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Conversely, procedural knowledge involves the ability to solve problems through the manipulation of mathematical skills with the help of pencil and paper, calculator, computer, and so forth see Figure 1. Obviously, mathematicians invented procedures based on mathematical concepts. The National Council of Teachers of Mathematics standards require that students know the procedures and understand their conceptual base.

Yet, there are two contrasting theories regarding the acquisition of these two types of knowledge. One is referred to as the conceptualchange view and the other as the empiricist view.


Conceptual and Procedural Knowledge: The Case of Mathematics

Figure 1. Without neglecting the importance of experience, the conceptual-change view defines learning as the modification of current concepts and emphasizes the role of concepts in the sense people make of their experience. This theory sheds light on what it would make sense to refer to as misconceptions, how those misconceptions develop, and what should be done to correct them.

References: Barr, C. Conceptual and procedural knowledge of mathematics: Does one lead to the other? Journal of Educational Psychology, 91 1 , Wolfram, C. Conrad Wolfram: Teaching kids real math with computers. Mewborn, P. Sztajn, D.

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