An Alternative Reconceptualization of Procedural and Conceptual Knowledge

After evaluating a proposal by Jon Star (2005) that mathematics educators reconceptualize the construct of procedural knowledge, we offer an alternative reconceptualization based on an extension and amalgamation of his and other conceptualizations of procedural knowledge and conceptual knowledge. We then propose some implications for research suggested by our critique and our alternative perspective.

In his November 2005 "Research Commentary" in this journal ("Reconceptualizing Procedural Knowledge"), Jon Star makes a valuable scholarly contribution by challenging mathematics educators to consider explicitly and carefully the constructs of procedural knowledge and conceptual knowledge.The heart of his argument, like that of de Jong and Ferguson-Hessler (1996), is that the popular use of these terms confounds knowledge categories or types (e.g., procedural knowledge) with knowledge properties or qualities, which can characterize any type of knowledge (e.g., degree of connectedness).STAR'S PROPOSAL FOR RECONCEPTUALIZING PROCEDURAL AND CONCEPTUAL KNOWLEDGE Star (2005) argues that each type of knowledge-procedural or conceptualcan have either a superficial or a deep quality.He explains the premise for proposing four distinct knowledge categories as follows (J.Star, personal communication, December 23, 2005): Preparation of this manuscript was supported, in part, by grants from the Spencer Foundation (2000400033-"Key Transitions in Preschoolers' Number and Arithmetic Development: The Psychological Foundations of Early Childhood Mathematics Education), the Institute of Education Science (R305K050082-"Developing an Intervention to Foster Early Number Sense and Skill"), and the National Institutes of Health (R01-HD051538-01-"Computer-guided Comprehensive Mathematics Assessment for Young Children").The opinions expressed are solely those of the authors and do not necessarily reflect the position, policy, or endorsement of the Spencer Foundation, the Institute of Education Science, or the National Institutes of Health.
In the absence of rich connections, knowledge is referred to as procedural by many if not most mathematics educators. . . .The adjectives "conceptual" and "procedural" are typically used to refer to the [number] of connections (knowledge quality . ..), with "conceptual" meaning richly connected.The use of these terms to refer to a type of knowledge . . . is largely lost.In common usage, procedural knowledge does not mean knowledge of procedures, but instead it refers to knowledge that is not richly connected.
In his 2005 commentary, Star concluded that equating procedural knowledge with superficial, sparsely connected, or rote knowledge and conceptual knowledge with deep, richly connected, and meaningful knowledge "makes it difficult to consider (or even name) the knowledge that belongs in the deep procedural" or superficial conceptual categories (p. 408).This confusion also makes it "difficult to speak of the kind of knowledge that underlies flexible use of procedures" (J.Star, personal communication, December 23, 2005).Star (2005) noted that flexibility, a key "indicator of deep procedural knowledge," is "often overlooked" in mathematics education research and not "even accounted for in typical definitions of conceptual and procedural knowledge" (p.409).
Like de Jong and Ferguson-Hessler (1996), Star (2005) argued that knowledge type and quality should be treated as independent dimensions.To disentangle the two, he proposed defining conceptual knowledge as "knowledge of concepts or principles"-as knowledge that involves relations or connections (but not necessarily rich ones).He defined procedural knowledge as "knowledge of procedures" and deep procedural knowledge as involving "comprehension, flexibility, and critical judgment and [as] distinct from (but possibly related to) knowledge of concepts" (p.408).

Mathematics Education Researchers' Views
If Hiebert and Lefevre's (1986) initial definition of conceptual knowledge and aspects of their characterization of procedural knowledge are considered in isolation, then Star (2005) is correct in saying that they confound knowledge type and quality.Hiebert and Lefevre defined the former "as knowledge that is rich in relationships" and that cannot exist as "an isolated piece of information" (pp.3-4; see also Gray & Tall, 1994;Hiebert & Wearne, 1986).In contrast to conceptual knowledge, which is saturated with relationships of many kinds, Hiebert and Lefevre characterized procedural knowledge as involving primarily sequential relations.Subsequently, some mathematics educators, including the first author of this commentary, have indeed been guilty of oversimplifying their claims and loosely or inadvertently equating "knowledge memorized by rote . . .with computational skill or procedural knowledge" (Baroody, 2003, p. 4).
MERs have attempted to define and study both deep procedural knowledge and superficial conceptual knowledge.They have long contrasted unconnected, disembodied, meaningless, context-bound, or mechanical procedures (what could be called a "weak scheme") with well-connected, contextualized, integrated, meaningful, general, or strategic procedural knowledge (what could be called a "strong scheme"; see row A of Table 1).Analogously, well-connected conceptual knowledge has been contrasted with sparsely connected conceptual knowledge (see row B of Table 1).For instance, strong schemas-which involve generalizations broad in scope, high standards of internal (logical) consistency, principle-driven comprehension, and principled bases for a priori reasoning (i.e., predictions are derived logically)-have been proposed to underlie deep conceptual knowledge.Weak schemas-which entail generalizations local in scope, low standards of internal (logical) consistency, precedent-driven comprehension, and no logical basis for a priori reasoning (i.e., predictions are looked up)-have been posited to underlie superficial conceptual knowledge and to explain why younger children's concepts may be less deep and sophisticated (e.g., less general, logical, interconnected, or flexible) than older children's or adults' (see row B of Table 1).
MERs have further offered constructs that apply to both procedural and conceptual knowledge and suggest that both superficial and deep forms of each share many characteristics (see row C of Table 1).For example, both superficial conceptual and procedural knowledge have been described as routine expertise (knowledge that can be applied effectively to familiar, but not new, tasks), and both deep conceptual and procedural knowledge have been characterized as adaptive expertise (meaningful knowledge that can be applied creatively, flexibly, and appropriately to new, as well as familiar, tasks).
Consider children who memorize the names of different classes of two-dimensional shapes (e.g., triangle, square, rectangle and circle) by rote.MERs who narrowly interpret Hiebert and Lefevre (1986) might define such unconnected knowledge as procedural (as knowledge of the surface features of mathematical representations).Probably most, though, would define it as unconnected or sparsely connected concepts-as van Hiele's (1986) Level 1 (visualization).Consistent with a weak schema, Level 1 thinkers use appearances and prototypical examples (a precedent) to mechanically recognize, classify, and name shapes (Clements, Hatano, 1988, 2003Swaminathan, Hannibal, & Sarama, 1999).For example, a ᮀ is a square and not a rectangle "because it looks like a square, not like a rectangle."In contrast, Level 2 understanding (informal deduction) entails the use of well-connected knowledge (e.g., how the attributes of concepts are interrelated) to recognize that a square is a special kind of rectangle.
In brief, although some MERs may exclusively equate sparsely connected knowledge with procedural knowledge and richly connected knowledge with conceptual knowledge, probably most do not.Nevertheless, Star's (2005) point that mathematics educators do not have an entirely uniform and clear-cut definition of procedural and conceptual knowledge is valid.Star (2005) does not clearly disentangle knowledge type and quality.He cites as an example of deep procedural knowledge vanLehn's (1990) construct of "teleological understanding of a procedure" (e.g., comprehension of a procedure's design or justification for its use).Knowledge about procedures, however, can be either superficial (e.g., permit a recitation of the steps in order) or deep (e.g., empower a cogent explanation of how the steps are interrelated to achieve a goal).Although conceptual knowledge is not necessary for the former, it is unclear how substantially deep comprehension of a procedure can exist without understanding its rationale (e.g., the conceptual basis for each of its steps).Mathematical procedures are not developed in a vacuum.They are created to solve a problem, and their steps are worked out according to mathematical principles and logic.Likewise, deep conceptual knowledge depends on knowing the tools for applying and extending mathematical ideas (Kilpatrick, Swafford, & Findell, 2001).Thus, although an effort to separate quality and type of knowledge makes sense in theory, psychologically speaking, deep procedural and conceptual knowledge cannot be separated.Star (2005) suggests defining concept as psychologists do, as an equivalence class (Piaget, 1965;but cf. Davis, 1983).For example, learning the concept of dog entails inducing the attributes common to all members of that class (Mix, Sandhofer, & Baroody, 2005).As Star allows, however, such a definition implies connected knowledge, although not necessarily rich connections.Although his characterization of conceptual knowledge differs from that given initially by Hiebert and Lefevre (1986), it does not differ from that of MERs who view conceptual knowledge in terms of a continuum of connectedness-including Hiebert himself (see, e.g., Baroody, 1992;Hiebert & Carpenter, 1992;Lunkenbein, 1985;Resnick & Ford, 1981).Hiebert and Lefevre clearly imply that conceptual knowledge grows as additional connections are made via assimilation and integration.Hiebert and Wearne (1986) illustrate this point by noting that "the conceptual network of place value grows as . . .related bits of knowledge . . .are related to earlier ideas" (p.200).Because knowledge quality is defined by the number (and quality) of connections, completely disentangling knowledge quality and type would require defining conceptual knowledge in a way that excludes implying connections.
MERs widely agree that "flexibility of approach is the major cognitive requirement for solving nonroutine problems" (Kilpatrick et al., 2001, p. 127), that procedural fluency should entail flexible (as well as efficient and appropriate) application of procedures, and that each of these critical components of mathematical proficiency both benefits from and benefits conceptual understanding (see also, e.g., Carpenter et al., 1999;Davis, 1983;Gray & Tall, 1994;Ma, 1999;Romberg, Carpenter, & Kwako, 2005).MERs have characterized the integration of procedural and conceptual knowledge in various ways (see row C under the Deep column in Table 1).The construct of adaptive expertise, for one, unites the notions of deep conceptual knowledge, deep procedural knowledge, and flexibility (Baroody, 2003;Fuson, 2004).As Hatano (2003) noted, quoting page 15 of an article he wrote in 1982: I began by considering the following two questions: What kind of knowledge do flexible and inventive experts construct?And how do they construct it?In other words, I speculated about both the product of adaptive expertise and its acquisition process.Regarding the former, my attention focused on the conceptual knowledge underlying procedures."Flexibility and adaptability seem to be possible only when there is some corresponding conceptual knowledge to give meaning to each step of the skill and provide criteria for selection among alternative possibilities for each step within the procedures."(p.xi) Parenthetically, MERs have also considered what role other factors, such as beliefs, play in flexible problem solving (see, e.g., McLeod, 1992;Schoenfeld, 1985Schoenfeld, , 1992)).As an indicator of meaningful or deep procedural knowledge, MERs have, for some time, considered adaptive expertise or the flexible application of procedures to be critical in assessment (see, e.g., Baroody, 2003;Carpenter, 1986;Haapasalo, 2003;Hatano, 1988Hatano, , 2003;;Kilpatrick et al., 2001).Star (personal communication, December 23, 2005), however, does not consider existing efforts to define deep procedural knowledge to be adequate: Saying that flexible use of procedures [results from or is the byproduct of] the integration of conceptual and procedural knowledge (e.g., adaptive expertise) has bypassed a core issue about what procedural knowledge is, how it develops, and its role in the development of mathematical understanding.Procedural knowledge (or skill) is valuable in and of itself, not solely because of its connections with and integration to conceptual knowledge.
As Figure 1 illustrates, Star (2005) proposes a model in which procedural and conceptual knowledge are orthogonal, and in which deep, flexible procedural knowledge can be achieved with or without conceptual knowledge.That is, it can be achieved either with adaptive expertise (represented by cells A and, to some extent, B 1 and B 2 in Figure 1) or without it (i.e., largely through connections with nonconceptual knowledge, represented by cells B 3 and B 4 in Figure 1).
Existing evidence appears to support the conventional wisdom that, like procedural comprehension, substantial procedural flexibility and critical judgment require the integration of procedural knowledge with conceptual knowledge, not merely numerous connections with other aspects of nonconceptual knowledge (Carpenter, 1986;Carpenter, Levi, Franke, & Zeringue, 2005;Davis, 1983;Hatano, 2003).Wertheimer's (1945Wertheimer's ( /1959) observations of children who could not flexibly apply mechanically learned algorithms to even modestly novel problems have been cited by MERs to illustrate the need for connecting procedural knowledge with conceptual knowledge (Baroody, 2003;Resnick & Ford, 1981;Schoenfeld, 1986;Steffe, 1992).
It makes sense to question the conventional wisdom that integration with conceptual understanding is the sole source of procedural flexibility.Hatano (2003), for instance, may be overstating the case for integrated knowledge by claiming that some corresponding conceptual knowledge of each step of an algorithm is necessary for procedural adaptability.It is not accurate, however, to conclude that mathematics educators have overlooked flexibility or its basis in their theorizing or research.To stimulate further research on the relations among procedural knowledge, conceptual knowledge, and flexibility, we offer an alternative to Star's (2005) reconceptualization effort (see Figure 2).Consistent with his recommendation to define knowledge type independently of the degree of connectedness, we tentatively propose a modification of de Jong and Ferguson-Hessler's (1996) definitions, defining procedural knowledge as mental "actions or manipulations" (p.107), including rules, strategies, and algorithms, for completing a task, and defining conceptual knowledge as "knowledge about facts, [generalizations], and principles" (p.107).Like Star's model, but unlike de Jong and Ferguson-Hessler's, we-for the sake of simplicity-distinguish between only two types of knowledge.What de Jong and Ferguson-Hessler call situational knowledge (knowledge about problem situations or features that can provide a basis for representing a problem or seeking additional information) can be considered part of the other knowledge needed for a relatively complete and accurate grasp of procedural or conceptual knowledge (the horizontal axis in Figure 2).What they call strategic knowledge (general problem-solving strategies) can be viewed as procedural knowledge, if merely memorized by rote, or as the union of procedural and conceptual knowledge about problem solving, if meaningful.

Interdependence of Deep Procedural and Deep Conceptual Knowledge
A major difference between our perspective and Star's (2005) or de Jong and Ferguson-Hessler's (1996) proposal (as represented by cells B 3 and B 4 in Figure 1) is that, although (relatively) superficial procedural and conceptual knowledge may exist (relatively) independently, (relatively) deep procedural knowledge cannot exist without (relatively) deep conceptual knowledge or vice versa (Baroody, 2003).The degree of connectedness or mutual dependence between conceptual and procedural knowledge is represented by the horizontal axis in Figure 2.

Quality of Deep Procedural and Deep Conceptual Knowledge
Whereas Star (2005) seems to equate deep knowledge with richly connected knowledge only, de Jong and Ferguson-Hessler (1996) propose that knowledge quality can also include the level of structure (the degree of organization), abstractness (the generality or breadth of application), and accuracy.The vertical axis in Figure 2 depicts these other aspects of knowledge quality and other aspects of knowledge completeness, including connections within a knowledge type or to everyday situations and applications.
In our view, depth of understanding entails both the degree to which procedural and conceptual knowledge are interconnected and the extent to which that knowledge is otherwise complete, well structured, abstract, and accurate.In Figure 2 field until it expands) represents depth of understanding.Cell F represents superficial procedural knowledge (an isolated weak scheme); Cell E, superficial conceptual knowledge (an isolated weak schema)-a concept that is not understood.Cell A represents deep knowledge (an integrated strong schema and schemes).Cells D, C, and B represent progressively greater integration of procedural and conceptual knowledge with one another and with other aspects of knowledge (increasing adaptive expertise) and an increasingly well-structured, abstract, and accurate web of knowledge.

Big Ideas and Understanding Concepts and Procedures
Big ideas, which are overarching concepts that connect multiple concepts, procedures, or problems within or even across domains or topics (Baroody, Cibulsksis, Lai, & Li, 2004), are integral to achieving a deep understanding of both concepts and procedures.Big ideas serve a number of functions.
Provide a basis for understanding many concepts.For instance, the big idea of equal partitioning, or its informal analogy of fair sharing, is at the heart of comprehending the following concepts: 1. Unit principle.Unlike irrational numbers, any natural number can be expressed as the sum of units (e.g., 5 = 1 + 1 + 1 + 1 + 1), and any rational number can be expressed as the sum of unit fractions (e.g., 3/5 = 1/5 + 1/5 + 1/5).2.Even number.An even number of items can be shared fairly by exactly two people.3. Division.For example, an amount shared fairly among a certain number of people yields a share of what size for each person?4. Fraction.For example, 3/4 can be viewed as follows: Sharing three candy bars among four people, what is the size of each person's share? 5. Measurement.A continuous quantity such as length or area can be subdivided into equal size parts (units), which can then be counted.6. Mean.To find what a typical share size (score) would be if everyone had the same size share, combine all shares (scores), and then divvy up the total fairly among the number of people (number of scores).
Supply a rationale for various procedures.Big ideas can help children reinvent and comprehend standard algorithms or invent their own relatively efficient procedures.For instance, understanding division in terms of fair sharing can enable children to reinvent the long division algorithm or to invent other relatively efficient procedures to solve multidigit division problems (e.g., Ambrose et al., 2003).Such adaptive expertise can provide the basis for creative transfer-to everyday situations and to comprehending, discovering, or inventing more advanced concepts or procedures (Ambrose et al.;Baroody, 2003).
Relate diverse concepts (and connected procedures) within and among domains, and provide a basis for structuring or restructuring knowledge.The examples of equal partitioning previously discussed provide a connection among such diverse topics as number theory, operations on whole numbers, rational numbers, measurement, and statistics.Big ideas invite children to view mathematical knowledge as cohesive or structured rather than as a series of isolated procedures, definitions, and so forth.In particular, they invite students to look beyond surface features of procedures and concepts and see diverse aspects of knowledge as having the same underlying structure.In brief, big ideas seem integral to constructing well-connected, well-structured, abstract, and accurate knowledge-deep understanding of conceptual and procedural knowledge.

IMPLICATIONS FOR RESEARCH
Whether Star's (2005) perspective illustrated in Figure 1 or our alternative reconceptualization illustrated in Figure 2 is more accurate is an empirical question.A logical implication of his perspective is that flexible use of procedures can develop independently of conceptual knowledge, which would entail demonstrating that appreciable strategy choice or transfer knowledge can be fostered without, for example, understanding the rationale for the steps of procedures.
A logical implication of our alternative proposal is that at least a degree of conceptual knowledge is a necessary condition for (relatively) deep procedural knowledge (strategy choice or adaptive expertise with procedures in a domain) and vice versa.A corollary is the following revision of Hatano's conjecture (2003): Increasing integration with corresponding conceptual knowledge increases the accuracy, versatility, duration, and generality of strategy choice and adaptability.Put differently, procedural knowledge connected exclusively or largely with other nonconceptual knowledge (cells B 3 and B 4 in Figure 1) tends to yield more error-prone, rigid, short-term, or isolated extensions than would more conceptually connected procedural knowledge (Baroody, 2003;Carpenter, 1986).
A second implication is that building connections within a type of knowledge or even between types may be insufficient to maximize deep understanding; other qualities, such as well-structured knowledge that permits constructing a useful representation of the problem, need to be considered (Davis, 1983;Gray & Tall, 1994;Greeno, Riley, & Gelman, 1984;Haapasalo, 2003;Hähkiöniemi, 2006;Ma, 1999;Rittle-Johnson, Siegler, & Alibali, 2001).For example, Peled and Segalis (2005) found that abstracting procedural similarities and mapping similarities among domains was more effective than the former alone.Recent work on early algebra has begun to explore what kinds of connections while learning arithmetic provide a basis for learning algebra, as well as for generalizing arithmetic procedures and using them flexibly (Carpenter et al., 1999;Romberg et al., 2005).A key finding is that connections that can be explicitly recognized and justified are more powerful than those that cannot.The nature of the connections, then, does matter (T.Carpenter, personal communication, August 24, 2006).A corollary of this point is that connections or relations are not equal; some can be formed without understanding, others with some understanding, and yet others with considerable understanding (cf.Hiebert & Lefevre, 1986).
A third implication is that big ideas are a critical aspect of understanding the nature, structure, or quality of connections.Theoretically, they should provide connections with considerable understanding-a relatively strong and general basis for connecting procedures and concepts within and among domains and, thus, for promoting adaptive expertise (Baroody et al., 2004).Indeed, instruction based on big ideas-more than typical procedural, conceptual, or integrated procedural-conceptual instruction-may help facilitate a chain of self-directed and meaningful conceptual and procedural learning in new domains.For instance, a professional development program that focused on the big idea of algebra as a shorthand for generalizing and representing arithmetic patterns and relations (and justifying and using arithmetic generalizations) enabled participating teachers to help their students generate a wider range of meaningful strategies and better understand symbols, such as the equals sign, significantly better than nonparticipating teachers (Carpenter & Franke, 2001;Jacobs, Franke, Carpenter, Levi, & Battey, in press).
In conclusion, de Jong and Ferguson-Hessler's (1996) and Star's (2005) recommendation to disentangle knowledge type and quality-to define procedural and conceptual knowledge independently of the degree of connectedness-makes sense.It also makes both psychological and pedagogical sense, however, to view meaningful knowledge of mathematical procedures and concepts as intricately and necessarily interrelated, not as distinct categories of mathematics.A case can be made that conceptual understanding is a key basis for all other aspects of mathematical proficiency, including procedural fluency (Baroody, 2003), and that big ideas are particularly important in promoting all aspects of mathematical proficiency.Psychologists have long known that meaningful memorization is a more effective means than memorization by rote for ensuring retention and transfer of factual and procedural knowledge (Ginsburg, 1977;Katona, 1967;Skemp, 1978;Wertheimer, 1959).Linking procedural to conceptual knowledge can make learning facts and procedures easier, provide computational shortcuts, ensure fewer errors, and reduce forgetting (i.e., promote efficiency).Children who understand procedures are more likely to recognize real-world applications and be able to adjust their extant knowledge to new challenges or problems.This is, in part, why the National Research Council (Kilpatrick et al., 2001) recommended that the strands of mathematical proficiency be taught in an intertwined manner.

Figure 1 .
Figure 1.The orthogonal relation between procedural and conceptual knowledge proposed in Star's (2005) reconceptualized framework

Figure 2 .
Figure 2. The mutually dependent relation between procedural and conceptual knowledge suggested by a model of adaptive expertise.

Table 1
Efforts to Define Types and Qualities (Continua) of Procedural and Conceptual Knowledge