The Application and Development of an Addition Goal Sketch

Abstract

A schema based view of addition development is compared with Siegler's latest strategy-choice model, which includes an addition goal sketch (a basic understanding of "the goals and causal relations" of addition; Siegler & Crowley, 1994, p. 196). This metacognitive component in the latter model is presumed to develop as a child practices a basic counting based procedure for calculating sums and to subsequently affect strategy choice by identifying correct and incorrect procedures. A study of 20 kindergartners was undertaken to examine key assumptions of this model. Participants were individually interviewed to determine their own procedure use and their ability to evaluate correct and incorrect procedures. The results did not support a central prediction of the strategy-choice model-that children who possess a goal sketch should recognize as correct a relatively advanced (counting-on) procedure they themselves do not use. These results are consistent with the schema based view that accurately evaluating or learning such advanced procedures requires a qualitative leap in conceptual development. Theoretical, methodological, and instructional implications include a case for a simulation that models assimilation (a cognitive process underlying developmental readiness and conceptual development), a conceptually based taxonomy of addition development, and a nuanced instructional theory of addition (i.e., one that goes beyond the direct instruction vs. discovery learning debate).When and how do children construct the conceptual understanding necessary to evaluate and adopt new counting based procedures for determining sums? Once children construct a basic understanding of the purpose (goals) and nature (causal relations) of addition, does this goal sketch empower them to comprehend, assess, and learn any new counting based procedure, even relatively sophisticated or advanced procedures they themselves have never used? In instructional terms, is direct instruction (modeling by, e.g., a parent, early childhood teacher, older sibling, or television character) sufficient for learning all such procedures or accelerating addition development? Or, does a goal sketch enable children to assimilate, judge, and internalize basic procedures but not more advanced procedures that require a deeper conceptual understanding? That is, must educators take into account developmental readiness when teaching addition procedures and focus on promoting the conceptual underpinnings of more advanced procedures? The main purpose of this study is to address these theoretical and practical questions.Researchers generally agree that children use increasingly sophisticated and abstract counting based procedures to determine sums (e.g.

Spontaneous Usage of Different Shortcuts Based on the Commutativity Principle

“…We set aside the question whether concepts or strategies set the starting point in the development of mathematical concepts. Common to most current views [13,39–41] is the assumption is that procedural and conceptual knowledge develop iteratively with small increases in one leading to small increases in the other which in turn trigger new increases in the first. The endpoint of development should, in the best case scenario, be marked by an integrated concept.…”

“…Furthermore, behavior should reveal links between shortcuts based on the same concept, even when they are considerably different in procedure. Based on previous research [13–16] we expected that with increasing age, participant should adaptively select the shortcut strategy that best fits the shortcut options of the current task material [4] and usage of different shortcuts that are based on the same mathematical principle should increasingly correlate with age.…”

Why can't Johnny remember the basic facts?

“…Baroody et al [2006b] distinguished between two major categories of counting-based addition strategies. Concrete strategies entail representing one or both addends with objects before beginning the sum count.…”

“…Although concrete strategies are relatively easy to learn (either through self-invention or via direct instruction such as modeling), even among children with MD, more efficient abstract strategies are not, particularly among children with learning difficulties [e.g., Baroody, 1996Baroody, , 1999b. The key reason for this disjuncture is that concrete strategies require relatively basic informal knowledge to construct or learn, whereas abstract strategies necessitate relatively advanced conceptual knowledge of counting and numbers [Baroody et al, 2006b]. Not surprisingly, children from low-income families, who are more likely to have weak informal knowledge, tend to rely on less advanced or efficient counting strategies for determining sums [Baroody et al, 1982;.…”

“…Put differently, it is difficult to impose number sense on children. Baroody et al [2006b] noted that direct instruction can be effective when a child is developmentally ready and receptive and when the instruction clearly and directly builds on what the child knows. For example, modeling a concrete counting all strategy can be highly effective with kindergartners and children with serious MLD, because the strategy corresponds clearly with their informal concept of addition.…”