Conceptual knowledge in mathematics

A substantial number of studies regarding students learning of mathematics in general and above all calculus concepts involve two dimensions of knowledge- conceptual and procedural (See for instance, Engelbrecht et al., 2005; Hiebert & Lefevre, 1986; Lauritzen, 2012; Schneider & Stern, 2005; Star, 2005; Star & Stylianides, 2013). There are also scholars who use different terms to name the duality for instance, relational and instrumental (Skemp, 1976). In the more recent literature, the conceptual and procedural terms to name the duality are dominantly used (Star & Stylianides, 2013).

52 2.2.4.1. Conceptual knowledge

Conceptual knowledge is defined as the ability to demonstrate, interpret, and relate the verity of mathematical concepts correctly to a variety of problem-solving situations (Engelbrecht et al., 2005). Rittle-Johnson, Siegler, and Alibali (2001) define conceptual knowledge as a set of pieces of knowledge about a concept and skill of interconnecting these pieces into a whole or network. The essentials of these networks can be rules or procedures, and even problems given in various representations. One with conceptual knowledge in mathematics demonstrates the ability to decompose a given mathematical expression into pieces or express the network in verbal statements. Built-in to such knowledge is associated network of knowledge so that the whole is as important as the individual elements that connected to give the whole (Engelbrecht et al., 2005).

An influential theme that is common among several definitions of conceptual knowledge is, “making connection or relation.” The term “relational” has also used by Skemp (1976) to name one type of mathematical understanding as will be discussed later. This theme originated from the definition of conceptual knowledge given by Hiebert and Lefevre, which by itself is seen as a foundation for the subsequent definitions of conceptual knowledge (as in Star & Stylianides, 2013). Hiebert and Lefevre (1986, p. 3) define conceptual knowledge as “a type of knowledge that is loaded in associations”. It can be considered as an associated network of knowledge so that the whole is as important as the individual elements that connected to give the whole. Its connected nature promotes awareness and the ability to move from particular to general and flexibility during task performance.

According to Tall (2002), mathematical thinking is a cognitive composition that is friendly to the “biological structure of the human brain” (p. 16). It is massive store of knowledge and inner associations, which systematically deals with various cognitive tasks. This definition of mathematical thinking is more like the definition of conceptual knowledge by Hiebert and Lefevre (1986). In both definitions, the focus is not only the amount of knowledge available, but also the connection and integration among those pieces of knowledge. In Konicek-Moran and Keeley (2015) view, a student is

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said to have conceptual knowledge if she/he is able to- think with it, extend it to similar situations, verbalize it, and get a similar or different way of expressing it. Students use their conceptual knowledge to identify what and when to use definitions, rules, and procedures, and to distinction associated concepts and evaluate results (Schneider & Stern, 2005). It is accumulated in some forms of relational representations or hierarchies and is not attached to specific problem types, rather can be adapted to the different context of problems. It is rich in relationships or webs of correlated ideas and allows individuals to distinguish between these correlations (Lauritzen, 2012; Mahir, 2009). In addition, it can be easily verbalized, flexibly transformed in the course of deduction and reflection (Schneider & Stern, 2005).

2.2.4.2. Procedural knowledge

Procedural knowledge is commonly associated with knowledge of procedures, and the setting where the procedures can be executed (Star & Stylianides, 2013). Engelbrecht et al. (2005) define it as the ability to explain the solution to a problem via the exploitation of a set of rules and procedures that associated with algorithms and symbols. According to Rittle-Johnson et al. (2001), procedural knowledge is the ability to perform algorithms quickly and efficiently as a part of problem-solving. This knowledge type is attached to a specific problem type and therefore is not easy to generalize it to different arrangement of problems in the same domain.

Hiebert and Lefevre (1986) describe procedural knowledge in mathematics into two components. The first component involves being familiar with the language of mathematics which is the symbolic representation. The other component is the knowledge of rules and procedures of those symbols to solve problems. The main quality of the procedural knowledge is to be “executed in a predetermined linear sequence” (p. 6). Thus, procedural knowledge as compared to conceptual knowledge engages minimal cognitive awareness and a little cognitive resources. It is easy to learn, and it allows students to execute possible actions that could be properly performed to solve a given problem. Nevertheless, it is less connected and shallow in representation and hence hard to reflect and communicate (Schneider & Stern,

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2005). That is why Hiebert and Lefevre (1986, p. 6) emphasized that procedural knowledge is the “narrative of managing mathematical signs and syntaxes”. It requires only a consciousness of rules and not interpretations or analysis. However, this does not mean that procedural knowledge has no relevance. Rather, students must learn to master fundamental concepts and computation of procedures (Mahir, 2009; Schoenfeld, 1992). Each one is quite limited unless it is connected to the other (Lauritzen, 2012; Rittle-Johnson et al., 2001).

2.2.4.3. Relational understanding

Skemp‟s relational understanding refers to both the ability to perform procedures and to justify why those procedures and rules are used, whereas, instrumental understanding represents knowing the rules, and procedures of mathematics. He argues that in the short run the later may be more pleasing because learning how to do something is usually easier to memorize than learning something with deep meaning attached and then relating that to how it works. Moreover, even for teachers, instrumental understanding is easier to make assessment than relational understanding. In the long run, however, relational understanding is more helpful. With regard to retention period of knowledge as mentioned above, Crowley (2000) as in Tall (2002, p. 16) comments that even average ability student works in a “cognitive kit-bag” that lack connection and perform explicit procedures. Resulting in the spot success and satisfaction and possibly, “long-term cognitive load and failure”. Instrumental oriented students can be identified from their performance in classroom tasks. Those students can perform simple routine exercises very well, but stack for items that are different in nature from the usual classroom and textbook items (Gray & Tall, 1994). Thus, Skemp strongly argues that teaching should promote relational understanding.

Going back to the conceptual and procedural duality of knowledge, in the more recent research literature, Skemp‟s instrumental and relational understanding referred to procedural and conceptual knowledge respectively (Wangle, 2013). Thus, in this study, too conceptual understanding and relational understanding is considered synonymous.

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2.2.4.4. Contextual definition of conceptual Knowledge

Star (2005) points out that the term conceptual knowledge includes both what is known, and the way that it can be built-in. Likewise, the term procedural knowledge specifies knowledge of procedures and the way that procedures can be known. In conceptual knowledge, the construction can be deep and rich in the association of the networks, whereas, in the procedural knowledge it is shallow and less in connection. Thus, Star argues that the description of knowledge dually like this encompasses both knowledge types and knowledge quality. These two aspects of knowledge and the interplay between them is presented in Table 2 taken from Star (2005, p. 408).

Table 2: Types and qualities of procedural and conceptual knowledge Knowledge

type

Knowledge quality

Superficial Deep

Procedural

Common usage of procedural knowledge

?

Conceptual ?

Common usage of conceptual knowledge

Source: Star 2005, p. 408.

Star further argues that the present practice on the duality of knowledge makes it hard to think and denote the knowledge that is deep in quality and procedural in type. Duffin and Simpson (2000) describe, “Depth of understanding” as the ability to explain and justify each step of a problem-solving in mathematical terms. While the surface level procedural knowledge is automated skills on ordinary rules of algorithms, the deep level serves the purpose of creating and modifying the superficial level. Thus, deep procedural knowledge is as important as conceptual knowledge.

The intention of the researcher in this study is not to claim that conceptual knowledge is more essential than procedural knowledge. He strongly believes both are important

Genuine conceptual Knowledge

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aspects of students‟ knowledge. Therefore, both deserve careful attention. With regard to the significance of both types of mathematical expertise, Hiebert and Lefevre‟s (1986) comment that: what makes a mathematical knowledge complete is not only the existence of both types of knowledge but also the strength of the integration among them. When both exist but lack integration, students may show interest and initiation to participate in problem-solving but remain unsuccessful.

In this study, conceptual knowledge refers to knowledge of both concepts and procedures, which is integrated and deep in quality. In particular, conceptual knowledge about a mathematical concept consists of the knowledge to compute procedures and to justify the reasoning employed within relevant representation forms together with the ability to communicate in written, in a coherent, consistent, and flexible mathematical practice. According to Star, “types and qualities” description of the duality, this definition refers to the area where the deep in procedural, and deep in conceptual overlaps. Defined in this way, conceptual knowledge deserves the description that it is an adequate competence to solve all types of problems and tasks. The next section presents constructs that are the manifestations of this conceptual knowledge in calculus.