Many authors have attempted to discern and classify types of knowledge. Indeed, the history of these attempts stretches back to antiquity and will not be covered here. It must be emphasized that what is considered knowledge is not universal and unchanging; even what we take as scientific knowledge has its roots in 20th century philosophy. We focus our attention on those knowledge classifications that have appeared in the education, and related, literature.
The primary focus of this working group was on procedural knowledge, which is often defined alongside conceptual knowledge. Perhaps the most commonly accepted definitions of procedural and conceptual knowledge are due to Hiebert and Lefevre (1986):
[Conceptual knowledge is] 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 terms of procedural knowledge:
One kind of procedural knowledge is a familiarity with the individual symbols of the
system and with the syntactic conventions for acceptable configurations of symbols.
The second kind of procedural knowledge consists of rules or procedures for solving
mathematical problems. Many of the procedures that students possess probably are
chains of prescriptions for manipulating symbols.
The connotations are clear: conceptual knowledge is somehow ‘better’ than procedural knowledge. This has led some authors to expand on these basic definitions, considering not just type of knowledge but depth. de Jong and Ferguson-Hessler (1996) review some of the knowledge constructs present in the literature and attempt to synthesize them. In the literature they review, they locate the following types of knowledge: generic (or general) and domain specific, concrete and abstract, formal and informal, declarative and proceduralized, conceptual and procedural, elaborated and compiled, unstructured and (highly) structured,
tacit or inert, strategic, ‘knowledge acquisition’, situated, and meta knowledge. Understandably, this plethora of knowledge types can cause confusion for both researchers and practitioners. In an attempt to consolidate these constructs to “avoid the introduction of still more types of knowledge that do nothing more than to describe properties of generally accepted types of knowledge” (p. 105), the authors place the definitions in a two-dimensional array. One axis is type, the other, quality. Working from their field, physics, the authors
identify four distinct types of knowledge, two of which — conceptual and procedural — are relevant to the current discussion. As for quality, the authors propose deep, associated with comprehension and abstraction, and critical judgement and evaluation, and surface, associated with reproduction and rote learning, trial and error, and a lack of critical judgement. This way of classifying knowledge has lead Star (2002) to enquire about deep procedural knowledge.
In a sense, the question Star (2002) poses is natural: we know what deep conceptual and superficial procedural knowledge look like, but what is deep procedural knowledge? Part of the problem in understanding deep procedural knowledge is we seldom look for it. Conceptual knowledge is measured verbally and through a variety of tasks, while procedures are measured in terms of task completion – is the answer correct or not? This binary assessment obscures the richness present in carrying out a procedure. Another issue is that mathematics education literature tends to draw from the same well of problems, those found in primary or early-secondary school. What are absent are higher-level problems where the richness in procedure execution is more transparent (Star, 2002). Any derivative in a calculus course, for example, can be computed in a great number of ways, all of which, if done correctly, will yield the same expression. It is while performing an operation like this that we encounter a rich mathematical performance by the student.
It is natural to assume that the more one understands about a certain concept, addition, say, the easier it will be to perform an operation based on this concept. Although this may be the case for some concepts, it is almost certainly not a general phenomenon. This assumption is brought about by what Star (2002) identifies as a deficit of the mathematics education literature: the concepts typically examined are taken from primary and secondary school. Examples of higher grade-level concepts are needed to illustrate the dynamic interplay
between knowing and doing. A good place to start is differentiation (Maciejewski & Mamolo, 2011). The derivative of a function is defined as the limit of a quotient of functions. There are many ways to conceptualize a derivative – as a rate of change, the slope of a tangent line, etc. – but no matter how deeply the concept of derivative is understood, the procedure for finding the derivative of a function remains opaque. Not only this, but the inability to compute derivatives inhibits how deeply the concept can be known. Furthermore, the acquisition of neither concepts nor procedures necessarily precedes the other (Rittle-Johnson & Siegler, 1998). A few studies have supported the notion that concepts and procedures can play off of each other, one reinforcing and strengthening the other (Byrnes & Wasik, 1991; Rittle- Johnson & Siegler, 1998). This may have profound implications for teaching practices: perhaps it is better to start with procedures with some students, concepts with others.
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