Conceptual Knowledge Vs Procedural Knowledge Analysis

Although conceptual and procedural knowledge cannot always be separated, it is useful to distinguish between the two types of knowledge to better understand knowledge development.
First consider conceptual knowledge. A concept is ‘an abstract or generic idea generalized from particular instances’ (Merriam-Webster’s Collegiate Dictionary, 2012). Knowledge of concepts is often referred to as conceptual knowledge (e.g. Byrnes & Wasik, 1991; Canobi, 2009; Rittle-Johnson, Siegler, & Alibali, 2001). This knowledge is usually not tied to particular problem types. It can be implicit or explicit, and thus does not have to be verbalizable (e.g. Goldin Meadow, Alibali, & Church, 1993). The National Research Council adopted a similar definition in its
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Star (2005) noted that: ‘The term conceptual knowledge has come to encompass not only what is known (knowledge of concepts) but also one way that concepts can be known (e.g. deeply and with rich connections)’ (p. 408). This definition is based on Hiebert and LeFevre’s definition in the seminal book edited by Hiebert (1986):
‘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’ (pp. 3–4).
After interviewing a number of mathematics education researchers, Baroody and colleagues (Baroody, Feil, & Johnson, 2007) suggested that conceptual knowledge should be defined as ‘knowledge about facts, [generalizations], and principles’ (p. 107), without
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Within psychology, particularly in computational models, there has sometimes been the additional constraint that procedural knowledge is implicit knowledge that cannot be verbalized directly. For example, John Anderson (1993) claimed: ‘procedural knowledge is knowledge people can only manifest in their performance …. procedural knowledge is not reportable’ (pp. 18, 21). Although later accounts of explicit and implicit knowledge in ACT-R (Adaptive Control of Thought—Rational) (Lebiere, Wallach, & Taatgen, 1998; Taatgen, 1999) do not repeat this claim, Sun, Merrill, and Peterson (2001) concluded that: ‘The inaccessibility of procedural knowledge is accepted by most researchers and embodied in most computational models that capture procedural skills’ (p. 206). In part, this is because the models are often of procedural knowledge that has been automatized through extensive practice. However, at least in mathematical problem solving, people often know and use procedures that are not automatized, but rather require conscious selection, reflection, and sequencing of steps (e.g. solving complex algebraic equations), and this knowledge of procedures can be verbalized (e.g. Star & Newton,

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