Patterns of knowledge in children's addition.

Abstract

Patterns of conceptual and procedural knowledge of addition were examined in 5- to 8-year-olds (N = 80). Conceptual knowledge was measured by assessing children's responses to problems in which addends were reordered or decomposed and recombined. Problems were presented using abstract symbols, numbers, and physical objects. Children were more successful in noticing that addends had been reordered rather than decomposed and in noticing the decomposition of addends presented with objects rather than with symbols. Distinct profiles of procedural competence were derived from an analysis of children's problem-solving accuracy and strategies. Profiles were associated with patterns of conceptual knowledge and with age, although age and conceptual knowledge were not related. Findings highlight the usefulness of identifying profiles of procedural and conceptual knowledge for understanding the development of children's knowledge of addition.

How we use what we learn in Math: An integrative account of the development of commutativity.

“…Nevertheless, it does not fit the general claim that with experience, children become faster and more accurate at solving addition problems and also tend to use more sophisticated strategies, such as order-irrelevant, decomposition, and retrieval strategies (Baroody et al, 1983;Canobi et al, 1998Canobi et al, , 20022003;Geary, Brown & Samaranayake, 1991;Goldman, Mertz & Pellegrino, 1989;Resnick, 1992;Rittle-Johnson & Siegler, 1998;Siegler, 1987;but see, McNeil, 2007;Robinson & Dubé, 2009;Robinson & Ninowski, 2003;Torbeyns et al, 2009). It also seems to contradict the results of Baroody et al (1983), which show that approximately 80% of their third graders applied the commutativity-based shortcut to solve arithmetic problems (see also, Canobi et al, 2003).…”

“…One frequently used approach to measure procedural knowledge is to ask children to solve addition problems and afterwards have them explain their strategies (e.g., Baroody & Gannon, 1984;Baroody, Ginsburg & Waxman, 1983;Bisanz & LeFevre, 1992;Canobi et al, 1998Canobi et al, , 2002Canobi et al, , 2003Cowan & Renton, 1996). Conceptual knowledge in these studies has, for example, been assessed by letting children observe a puppet solving problem pairs (see, e.g.…”

Concept–procedure interactions in children’s addition and subtraction

“…Despite strong relations between schoolchildren's conceptual understanding and their reported problem-solving procedures and accuracy (Canobi, 2004(Canobi, , 2005Canobi et al, 1998Canobi et al, , 2003, the nature of concept-procedure interactions in basic addition and subtraction is not yet well understood. For example, cross-sectional research indicates that children who correctly apply concepts such as inversion and commutativity to related problems solve randomly ordered problems with greater accuracy than do other children (Canobi, 2004(Canobi, , 2005Canobi et al, 1998Canobi et al, , 2003.…”

The Degree of Abstraction in Solving Addition and Subtraction Problems

“…These levels follow a progressive order of comprehension in the child (Canobi, Reeve, & Pattison, 2003;Fuson, Smith, & LoCicero, 1997). This was confirmed by Maccini and Hughes (2000) with a teaching sequence of three levels: concrete (manipulation of physical objects), semi-concrete (representation with drawings), and abstract (use of mathematical symbols).…”