Relations Between Conceptual and Procedural Knowledge

Throughout this century, instructional reform has oscillated between emphasizing mastering of facts and procedures on the one hand and emphasizing understanding of concepts on the other (Hiebert & Lefevre, 1986). Few today would argue that either type of mathematical knowledge should be taught to the exclusion of the other. Much less agreement exists, however, concerning the balance between the two that should be pursued or concerning how to design instruction that will inculcate both types of knowledge. Multidigit addition and subtraction has proved to be an especially fruitful domain for studying the relations between conceptual and procedural knowledge. Children spend several years learning multidigit arithmetic. They must learn the carrying procedure for addition and the borrowing procedure for subtraction. Understanding these procedures requires understanding of the concept of place, that each position in a multidigit number represents a successively higher power of ten. It also requires understanding that a multidigit number can be represented in different ways, for example, 23 can be represented as 1 "10" and 13 "1's".

Many children have difficulty understanding place value, and, as noted earlier, they frequently use buggy procedures that reflect this lack of understanding. For example, second-graders often do not correctly carry when adding multidigit numbers (Fuson & Briars, 1990). Instead, they either write the two-digit sums beneath each column of single-digit addends (e. g., 568+778=121316) or ignore the carried values (e. g.,

568+778=1236). Although there are exceptions, procedural skill and conceptual understanding usually are highly correlated. One source of evidence for this view is cross-national studies. For example, comparisons of Korean and American elementary school children have revealed parallel national differences in conceptual and procedural knowledge of multidigit addition and subtraction. Fuson and Kwon (1992) asked Korean second and third graders to solve two and three digit addition and subtraction problems that require carrying or regrouping. Then the children were presented several measures of conceptual understanding: ability to identify correctly and incorrectly worked out addition and subtraction problems, to explain the basis of the correct procedure, and to indicate the place value of digits within a number. Almost all of the Korean children used correct procedures to solve the problems and also succeeded on all of the measures of conceptual understanding. Stevenson and Stigler (1992) reported similar procedural and conceptual competence in first through fifth graders in Japan and China.

On the other hand, a number of studies reviewed in Fuson (1990) indicated that American children, ranging from second to fifth grade, frequently lack both conceptual and procedural knowledge of multidigit addition and subtraction. Lack of conceptual understanding was evident in findings that almost half of third graders incorrectly identified the place value of digits within multidigit numbers (Kouba, Carpenter & Swafford, 1989; Labinowicz, 1985), and in findings that most second through fifth graders could not demonstrate or explain ten-for-one trading with concrete representations (Ross, 1986). Lack of procedural knowledge was evident in findings that children of these ages frequently erred while using paper and pencil to solve multidigit addition and subtraction problems (Brown & Burton, 1978; Fuson & Briars, 1990; Kouba et al., 1989; Labinowicz, 1985; Stevenson and Stigler, 1992).

 Taken together, these results suggest that conceptual and procedural knowledge are related; Asian children have both, and American children lack both. Within the U. S., conceptual and procedural competence are also highly correlated. Second and third graders who correctly execute the subtraction borrowing procedure also are more accurate in detecting conceptual flaws in a puppet’s subtraction procedures than are children who do not execute the subtraction algorithm consistently corrrectly (Cauley, 1988). Conceptual understanding of multidigit addition and subtraction and the ability to invent effective computational procedures are also positively correlated in first through fourth graders (Hiebert & Wearne, 1996). This correlation leaves open the possibility that conceptual understanding could be causally related to children inventing adequate computational procedures, but also the possibility that knowing the correct procedure could be causally related to increased conceptual understanding (by allowing children to reflect on why the correct procedure is correct.) One relevant source of evidence is examination of the order in which individual children gain procedural and conceptual competence. It turns out that a substantial percentage of children first gain conceptual understanding and then procedural competence, but that another substantial percentage do the opposite (Hiebert & Wearne, 1996).

Studies aimed at improving teaching of multidigit addition and subtraction typically emphasize linking steps in the procedures to the concepts that support them. In general, these teaching techniques successfully increase both conceptual and procedural knowledge. Although not currently conclusive, they suggest that instruction that emphasizes conceptual understanding as well as procedural skill is more effective in building both kinds of competence than instruction that only focuses on procedural skill (Fuson & Briars, 1990; Hiebert & Wearne, 1996). A question that remains, however, is which type of knowledge should be emphasized first. Many opinions have been offered on this topic, but until recently, no directly-relevant experimental evidence was available. A recent study by Rittle- Johnson and Alibali (in press), however, provides such evidence. They examined fifth graders’ performance on mathematical equality problems of the form a+b+c=_____+c. Some randomly-selected children were presented conceptually-oriented instruction, other children were presented procedurally-oriented instruction, and yet others were presented neither. Then all children were presented practice solving problems, followed by a posttest that tested both conceptual and procedural knowledge. The conceptually-oriented instruction produced substantial gains in both kinds of knowledge; the procedurally-oriented instruction produced substantial gains in procedural knowledge and smaller gains in conceptual knowledge. To the degree that this result proves general, it suggests that conceptual instruction should be undertaken before instruction aimed at teaching procedures.