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Journal Article

Children's Understanding of Numerical Variables.

01 Sep 1978-Mathematics in School-Vol. 7, Iss: 4, pp 23-26
About: This article is published in Mathematics in School.The article was published on 1978-09-01 and is currently open access. It has received 144 citations till now. The article focuses on the topics: Educational research & Concept learning.
Citations
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Journal ArticleDOI
TL;DR: This article investigated the upper limits of the students' informal processes in the solution of first degree equations in one unknown prior to any instruction and found that the existence of acognitive gap between arithmetic and algebra, a cognitive gap that can be characterized asthe students' inability to operate spontaneously with or on the unknown.
Abstract: Serious attempts are being made to improve the students' preparation for algebra. However, without a clear-cut demarcation between arithmetic and algebra, most of these undertakings merely provide either an earlier introduction of the topic or simply spread it out over a longer period of instruction. The present study investigates the upper limits of the students' informal processes in the solution of first degree equations in one unknown prior to any instruction. The results indicate the existence of acognitive gap between arithmetic and algebra, a cognitive gap that can be characterized asthe students' inability to operate spontaneously with or on the unknown. Furthermore, the study reveals other difficulties of a pre-algebraic nature such as a tendency to detach a numeral from the preceding minus sign in the grouping of numerical terms and problems in the acceptance of the equal symbol to denote a decomposition into a difference as in 23=37−n which leads some students to read such equations from right to left.

359 citations

Journal ArticleDOI
TL;DR: The authors found that a large proportion of science-oriented college students were unable to solve a very simple kind of algebra word problem, and several thinking-aloud protocols were presented in order to formulate hypotheses for why the errors occurred, examine the phenomenon of shifts between inconsistent approaches within a single solution, and illustrate a method of diagramming thought processes.
Abstract: This article describes test data showing that a large proportion of science-oriented college students were unable to solve a very simple kind of algebra word problem. Several thinking-aloud protocols are presented in order to (a) formulate hypotheses for why the errors occurred, (b) examine the phenomenon of shifts between inconsistent approaches within a single solution, and (c) illustrate a method of diagramming thought processes. The data indicate that relatively advanced students can experience serious difficulties in symbolizing certain meaningful relationships with algebraic equations. Paige and Simon (1966) have shown that syntactic methods-that is, methods not dependent on comprehending the meaning of the described problem situation-are adequate for solving some algebra word problems in one variable. They point out, however, that these methods can produce incorrect or meaningless results in other problems. They also present evidence for the idea that some students use an internal "physical representation" that encodes qualitative features of a situation in solving word problems. A number of misconceptions concerning the meaning of algebraic equations have also been discussed by Wagner (Note 1), Collis (1978), Herscovics and Kieran (1980), Galvin and Bell (Note 2), Kiichemann (1978), and Matz (Note 3). Several of these researchers have noted a tendency on the part of some students to treat numerical variables as if they stood for objects rather than numbers. This study concentrates on one particular type of misconception in order to obtain a deeper understanding of its cognitive sources. Research attempting to describe thought processes in this area must be considered exploratory, since the processes involved are rather complex.

313 citations

Journal ArticleDOI
TL;DR: Examining both simple and more complex problems in two samples of college students provides empirical support for a trade-off between grounded, verbal representations, which show advantages on simpler problems, and abstract, symbolic representations, whose shows advantages on more complexblems.

183 citations


Cites background from "Children's Understanding of Numeric..."

  • ...One dimension along which representations vary is in how grounded or abstract they are (Paivio, 1986; Palmer, 1978)....

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  • ...…have identified student difficulties with algebraic symbols such as variables (Clement, 1982; Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Küchemann, 1978) and the equal sign (Kieran, 1981; McNeil & Alibali, 2004; RittleJohnson & Alibali, 1999), the difficulty of symbolic expressions…...

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Journal ArticleDOI
TL;DR: In a series of two in-vivo experiments, this article examined whether correct and incorrect examples with prompts for self-explanation can be effective for improving students' conceptual understanding and procedural skill in algebra when combined with guided practice.

160 citations


Cites background from "Children's Understanding of Numeric..."

  • ...…than of balance between the two sides (Baroody & Ginsburg, 1983; Kieran, 1981), that negative signs represent subtraction but do not modify the terms they precede (Vlassis, 2004), and that variables represent a single value (Booth, 1984; Knuth, Stephens, McNeil, & Alibali, 2006; Küchemann, 1978)....

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References
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Journal ArticleDOI
TL;DR: In this paper, the authors measured the proportion of children showing early and late concrete operational thinking and formal operational thinking in a sample of 10,000 children between the ages of 9 and 14.
Abstract: Summary. The proportion of children showing early and late concrete operational thinking, and early and late formal operational thinking was measured in a sample of 10,000 children between the ages of 9 and 14. The test instruments were a form of group test called Class Tasks, derived from the individual interview situations described by Piaget. These tests indicated that most children in early adolescence showed rapid development in concrete thinking, but that only one-fifth of the children showed the further development of formal operational thought. The representativeness of these findings was ensured by relating the distribution of Piagetian stages at each age-level to the norms of a standardised non-verbal reasoning test.

168 citations

Book ChapterDOI
01 Jan 1978
TL;DR: A Backward Look In a paper that appeared over 10 years ago (Lunzer, 1965), this author spoke of a desire to arrive at a clearer understanding of the kinds of advances in reasoning that appear as the child approaches adolescence as discussed by the authors.
Abstract: A Backward Look In a paper that appeared over 10 years ago (Lunzer, 1965), this author spoke of a desire to arrive at a clearer understanding of the kinds of advances in reasoning that appear as the child approaches adolescence. Are these advances sufficiently homogeneous and distinctive to warrant the use of the general term “formal reasoning” in opposition to the term “concrete reasoning” to characterize the achievements belonging to the years between six and nine? (p. 19)

25 citations