Addition and subtraction: A cognitive perspective

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    • "For the purposes of this study, only responses to addition or subtraction problems were recorded. Therefore, the task was reduced to 20 items: 6 change AWPs, 6 compare AWPs, 6 equalize AWPs, and 2 combine AWPs, based on the classification of Carpenter and Moser (1983; Cronbach's alpha reliability value for the 20 items is .95, and when only change, compare, and equalize problems are used this value is .94). "
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    ABSTRACT: Arithmetic word problem (AWP) solving is a highly demanding task for children with learning disabilities (LD) since verbal and mathematical information have to be integrated. This study examines specifically how syntactic awareness (SA), the ability to manage the grammatical structures of language, affects AWP solving. Three groups of children in elementary education were formed: children with arithmetic learning disabilities (ALD), children with reading learning disabilities (RLD), and children with comorbid arithmetic and reading learning disabilities (ARLD). Mediation analysis confirmed that SA was a mediator variable for both groups of children with reading disabilities when solving AWPs, but not for children in the ALD group. All groups performed below the control group in the problem solving task. When SA was controlled for, semantic structure and position of the unknown set were variables that affected both groups with ALD. Specifically, children with ALD only were more affected by the place of the unknown set.
    Journal of learning disabilities 02/2014; DOI:10.1177/0022219413520183 · 1.77 Impact Factor
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    • "To understand negatives, each of these aspects must be extended. For example, the educational literature recognized different senses to the '+' sign (Carpenter, Moser, & Romberg, 1982; Nesher & Katriel, 1977; Vergnaud, 1982): static (combine), dynamic (change), or comparing. Therefore, each of these senses needed to be extended to negatives. "
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    ABSTRACT: In this paper, we propose a new model for learning mathematical concepts which cannot develop informally. This is a middle-out approach in which the objects of the system can be mapped both to formal knowledge and to real-life problems. We hypothesize that the key process which governs the acquisition of knowledge within such learning systems is reasoning with mental models, which we hold to be similar to the acquisition of basic mathematical constructs in non-school settings. The paper reports the development of a system for learning negative numbers and a study in which two pairs of students were exposed to that system.
    01/2006: pages 286-293;
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    • "For the past thirty years researchers have been inquiring into what lies behind the major difficulties children encounter with word problems. At one stage, researchers made a distinction between different additive problems and classified them into three main categories: The research was conducted in several countries and all agreed on the same categorization of additive word problems (Nesher & Teubal, 1975; Nesher & Katriel, 1977; Carpenter, Moser, & Romberg, 1982; Vergnaud, 1982; Nesher, 1982a; Nesher, Greeno & Riley, 1982b; Greeno & Kintsch, 1985; Riley & Greeno, 1988; Vergnaud, 1988; Verschaffel, 1993; Kintsch, 1994). Table 1 presents the main categories of additive word problems agreed upon: "
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    ABSTRACT: This paper is about schemes and how they can assist in the learning of word problems in mathematics. First the paper presents the theoretical background and working definitions for schemes suggested by cognitive psychology. Analysis of various types of word problems and the developmental trend that children exhibit in solving these problems follows. A special analysis is devoted to a number of constrained schemes that underlie common mathematical word problem types. The use of schemes is extended to open-end word problems showing that these too can be better solved methodically with the help of schemes. Theoretical background This paper deals with schemes 1 and how schemes assist in learning arithmetic word problems. Many philosophers and psychologists considered the notion of schemes with some variations. The term 'scheme' is used as a means of perceiving the world as an innate logical development and as patterns of action. Piaget (Piaget & Inhelder, 1969; Piaget, 1970; Piaget, 1971, 1967; Piaget, 1985) dealt with schemes of action, writing: A schema of an action consists in those aspects which are repeatable, transposable, or generalisable (Piaget, 1980, p.205). Fischbein (Fischbein & Grossman, 1997; Fischbein 1999) based his definition on Piaget´s notion of a scheme which defines a scheme not merely as a perceptual framework, but rather as a pattern of action. Fischbein in particular believed that a scheme is also a strategy for solving a certain class of problems. He too, stressed the behavioral aspect of a scheme. For him a scheme is a plan for action. Let us take a simple example (a kinetic scheme of action): Opening a door by its handle, knowing that the handle is to be pushed down and then the door pushed in or pulled out, is a scheme. We hardly ever pay attention to it because for us it is instinctive, but once we enter another system which we do not recognize, such as in a train 1 We will use the term 'scheme' and 'schemes' (in English), although 'schema', 'schemata' (in Latin) and 'schemas' are used in quotations from other authors, and we regard all these to be interchangeable.
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