Conceptual and Procedural Knowledge

The Case of Mathematics

James Hiebert

• Publisher: Routledge
• Year: 2013
• eBook ISBN: 9781136559839
• Topic: Education
• Subtopic: Education Theory & Practice

First Published in 1986. This book is intended for those people who are interested in how mathematics is learned. It is intended especially for those who are interested in the mental processes involved in becoming mathematically competent and the mental processes that inhibit such competency from developing. The volume opens with an overview of the issue and then traces the relationships between conceptual and procedural knowledge in mathematics from preschool days through the years of formal schooling. Mathematics educators and cognitive psychologists from a variety of perspectives contribute theoretical arguments and empirical data to illuminate the nature of the relationships and, in tum, the nature of mathematics learning.

1	Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis

James Hiebert

Patricia Lefevre

University of Delaware

Conceptual and procedural knowledge of mathematics represents a distinction that has received a great deal of discussion and debate through the years. Questions of how students learn mathematics, and especially how they should be taught, turn on speculations about which type of knowledge is more important or what might be an appropriate balance between them. Additionally, discussions of conceptual and procedural knowledge extend beyond the boundaries of mathematics education. The distinction between concepts and procedures plays an important role in more general questions of knowledge acquisition. In some theories of learning and development, the distinction occupies center stage. Although the types of knowledge that are identified from theory to theory are not identical, there is much overlap. The differences are primarily in emphasis rather than kind. For example, Piaget (1978) distinguishes between conceptual understanding and successful action; Tulving (1983) distinguishes between semantic memory and episodic memory; Anderson (1983) distinguishes between declarative and procedural knowledge. Parallel distinctions are made in philosophical theories of knowledge. For example, Scheffler (1965) distinguishes between the propositional use of “knowing that” and the procedural use of “knowing how to.” The distinction between conceptual and procedural knowledge that we elaborate in this chapter is not synonymous with any of these distinctions, but it draws upon all of them.

Mathematics, with its tightly structured and clearly defined content, has provided an arena for much discussion of conceptual and procedural knowledge. Over the past century, considerations of these two kinds of mathematical knowledge have taken different forms using different labels. Probably the most widely recognized distinction has been that between skill and understanding. Often the discussions of skill and understanding have taken the form of a debate about which should receive greater emphasis during instruction. McLellan and Dewey (1895) argued for understanding and presented a mathematics curriculum they felt would raise the level of understanding beyond that existing in classrooms at the time. Thorndike (1922) presented the case for skill learning and described in detail how skills should be taught to maximize retention. Brownell (1935) opposed the emphasis on learning isolated skills and argued forcefully for an increased emphasis on understanding. Essentially the same debate was carried on periodically through Gagné’s (1977) emphasis on skill learning versus Bruner’s (1960) case for understanding. Along the way there have been many additional voices addressing the issue from a variety of perspectives (see Jones, 1970; Shulman, 1970).

Currently, cognitive psychologists and mathematics educators are looking again at conceptual and procedural knowledge in mathematics learning. Sometimes, the discussions are couched in terms different from those used in the past. For example, Resnick (1982) talks about semantics and syntax, and Gelman and Gallistel (1978) distinguish between principles and skills. Even within this volume, a variety of terms are used to differentiate between types of knowledge. Baroody and Ginsburg describe differences between meaningful and mechanical knowledge, and VanLehn distinguishes between schematic and teleologic knowledge. But, regardless of the labels, the division between types of knowledge lies in approximately the same place today as it has in the past.

There are, however, three important differences between current discussions of conceptual and procedural knowledge and the historic discussions of understanding and skill. First, the essays of the past have treated understandings and skills as instructional outcomes and have dealt with them in the context of advocating instructional programs. The issue has been whether skills, or understandings, or both should be emphasized during classroom instruction. The context for addressing the question of the relative importance of skills and understandings often has been the prescription of instructional programs. Today, many of the writings describe the acquisition of knowledge and the relationships between different kinds of knowledge. The implicit assumption is that more complete descriptions are a first step on the road to better prescriptions. Detailed descriptions are believed to provide a sound basis from which to develop effective instructional programs.

The second difference between past and current discussions of conceptual and procedural knowledge is found in the current attention to relationships between concepts and procedures. Historically, the two kinds of knowledge have been viewed as separate entities, sometimes competing for the teacher’s attention, at best coexisting as disjoint neighbors. Little interest has been shown in studying the relationships between the two kinds of knowledge. In contrast, there is a growing interest today in how concepts and procedures are related. Current discussions treat the two forms of knowledge as distinct, but linked in critical, mutually beneficial ways. The new language of cognitive science has facilitated this approach because a single language can now be used to deal with both forms of knowledge (Anderson, 1983; Davis, 1984; Norman & Rumelhart, 1975). It is no longer the case that different theories are needed to express the principles guiding the acquisition and application of each kind of knowledge; a single theoretical orientation can handle both conceptual and procedural knowledge.

The third difference between past and present discussions is that past distinctions between conceptual and procedural knowledge focused on mathematics learning in school, whereas recent discussions of the issue have broadened the scope to include preschool mathematics learning in informal settings. Although it has long been recognized that children enter school with significant mathematical competencies (Brownell, 1941; McLaughlin, 1935), it is only recently that these competencies have been analyzed in great detail. Some of the analyses have revealed that the distinction between conceptual and procedural knowledge is as appropriate and useful for understanding the acquisition of informal mathematics as for formal mathematics. Preschool children acquire certain procedures or skills along with concepts, understandings, or intuitions about mathematics. The relationships between these kinds of knowledge, even at this level, appears to be complex.

Although the recent orientation to the issue of conceptual and procedural knowledge promises to provide significant insights into mathematics learning and performance, the relationship between these forms of knowledge are not yet well understood. A primary reason for the intractable nature of the problem is that the types of knowledge themselves are difficult to define. The core of each is easy to describe, but the outside edges are hard to pin down.

Our position is that the distinction between conceptual and procedural knowledge is useful for thinking about mathematics learning, and the clearer we can be about the distinction, the better. We do not believe, however that the distinction provides a classification scheme into which all knowledge can or should be sorted. Not all knowledge can be usefully described as either conceptual or procedural. Some knowledge seems to be a little of both, and some knowledge seems to be neither. Nevertheless, we believe that it is possible to distinguish between the two types of knowledge and that such a distinction provides a way of interpreting the learning process that helps us better understand students’ failures and successes.

DEFINITIONS OF CONCEPTUAL AND PROCEDURAL KNOWLEDGE

Conceptual Knowledge

Conceptual knowledge is characterized most clearly as knowledge that is rich in relationships. It can be thought of as a connected web of knowledge, a network in which the linking relationships are as prominent as the discrete pieces of information. Relationships pervade the individual facts and propositions so that all pieces of information are linked to some network. In fact, a unit of conceptual knowledge cannot be an isolated piece of information; by definition it is a part of conceptual knowledge only if the holder recognizes its relationship to other pieces of information.

The development of conceptual knowledge is achieved by the construction of relationships between pieces of information. This linking process can occur between two pieces of information that already have been stored in memory or between an existing piece of knowledge and one that is newly learned. It may be helpful to consider each of these phenomena in turn. The literature of psychology and education is filled with accounts of insights gained when previously unrelated items are suddenly seen as related in some way. Such insights are the bases of discovery learning (Bruner, 1961). We characterize this as an increase in conceptual knowledge. Two illuminating accounts of this kind of conceptual knowledge growth in elementary mathematics are found in Ginsburg (1977) and Lawler (1981). Ginsburg describes many points in the learning of number and arithmetic where understanding involves building relationships between existing bits of knowledge. For example, Jane (age nine) understood multidigit subtraction for the first time when she recognized the connection between the algorithm she had memorized and her knowledge of the positional value of each digit (p. 155). Relationships can tie together small pieces of information or larger pieces that are themselves networks of sorts. When previously independent networks are related, there is a dramatic and significant cognitive reorganization (Lawler, 1981).

A second way in which conceptual knowledge grows is through the creation of relationships between existing knowledge and new information that is just entering the system. The example of Jane cited above would fit here if Jane had recognized the connection between algorithm and place value immediately upon being taught the algorithm. Again, this phenomenon has been described with a variety of labels. Perhaps “understanding” is the term used most often to describe the state of knowledge when new mathematical information is connected appropriately to existing knowledge (Davis, 1984; Skemp, 1971; Van Engen, 1953). Other terms, like “meaningful learning,” convey similar sentiments (Ausubel, 1967; Brownell, 1935; Greeno, 1983b). Regardless of the term used, the heart of the process involves assimilating (Piaget, 1960) the new material into appropriate knowledge networks or structures. The result is that the new material becomes part of an existing network.

It is useful to distinguish between two levels at which relationships between pieces of mathematical knowledge can be established. One level we will call primary. At this level the relationship connecting the information is constructed at the same level of abstractness (or at a less abstract level) than that at which the information itself is represented. That is, the relationship is no more abstract than the information it is connecting. The term abstract is used here to refer to the degree to which a unit of knowledge (or a relationship) is tied to specific contexts. Abstractness increases as knowledge becomes freed from specific contexts.

An example may help to clarify the idea of primary level relationships. When students learn about decimal numbers, they learn a variety of things about decimals, including the following two facts. First, the position values to the right of the decimal point are tenths, hundredths, and so on; second, when you add or subtract decimal numbers you line up the decimal points. Usually, it is expected that students will relate these two pieces of information and recognize that when you line up decimal points in addition you end up adding tenths with tenths, hundredths with hundredths, and so forth. If students do make the connection, they certainly have advanced their understanding of addition. But a noteworthy characteristic of this primary relationship is that it connects two pieces of information about decimal numbers and nothing more. It is tied to the decimal context.

Some relationships are constructed at a higher, more abstract level than the pieces of information they connect. We call this the reflective level. Relationships at this level are less tied to specific contexts. They often are created by recognizing similar core features in pieces of information that are superficially different. The relationships transcend the level at which the knowledge currently is represented, pull out the common features of different-looking pieces of knowledge, and tie them together. In the example cited earlier, the learner might step back mentally and recall that you line up numerals on the right to add whole numbers and end up adding units with units, tens with tens, hundreds with hundreds. When adding common fractions, you look for common denominators and end up adding the same size pieces together. Now the connection between the position value and lining up decimal points to add decimal numbers is recognized as a special case of the general idea that you always add things that are alike in some crucial way, things that have been measured with a common unit. Lining up decimal points results in adding together the parts of the decimal fractions that are the “same size.” This kind of a connection is at a reflective level because building it requires a process of stepping back and reflecting on the information being connected. It is at a higher level than the primary level, because from its vantage point the learner can see much more of the mathematical terrain.

There are other ways to describe the different kinds of relationships that are part of one’s conceptual knowledge in mathematics, but the primary and reflective levels provide a useful distinction. The analysis is similar in some important ways to the different types of understanding described by Greeno (1980, 1983b) and the different types of intelligence proposed by Skemp (1971). Although this distinction is not always made explicit in the remaining discussion, it is important to remember that not all relationships are of a single kind.

Procedural Knowledge

Procedural knowledge, as we define it here, is made up of two distinct parts. One part is composed of the formal language, or symbol representation system, of mathematics. The other part consists of the algorithms, or rules, for completing mathematical tasks. The first part is sometimes called the “form” of mathematics (Byers & Erlwanger, 1984). It includes a familiarity with the symbols used to represent mathematical ideas and an awareness of the syntactic rules for writing symbols in an acceptable form. For example, those who possess this aspect of procedural knowledge would recognize that the expression 3.5 ÷

= 2.71 is syntactically acceptable (although they may not know the “answer”) and that 6 + =

2 is not acceptable. At more advanced levels of mathematics, knowledge of form includes knowledge of the syntactic configurations of formal proofs. This does not include the content or logic of proofs, only the style in which proof statements are written. Notice that, in general, knowledge of the symbols and syntax of mathematics implies only an awareness of surface features, not a knowledge of meani...

Table of contents

• Cover
• Halftitle
• Title
• Copyright
• Contents
• Preface
• Foreword
• 1 Conceptual and Procedural Knowledge in Mathematics: An Introductory Analysis
• 2 The Notion of Principle: The Case of Counting
• 3 Children’s Mastery of Written Numerals and the Construction of Basic Number Concepts
• 4 The Relationship Between Initial Meaningful and Mechanical Knowledge of Arithmetic
• 5 Conceptual Knowledge as a Foundation for Procedural Knowledge: Implications from Research on the Initial Learning of Arithmetic
• 6 Arithmetic Procedures are Induced from Examples
• 7 Using Conceptual and Procedural Knowledge: A Focus on Relationships
• 8 Procedures Over Concepts: The Acquisition of Decimal Number Knowledge
• 9 On Having and Using Geometric Knowledge
• 10 Conceptual and Procedural Knowledge in Mathematics: A Summary Analysis
• Author Index
• Subject Index