Modeling and the Initiation into Algebra

Janvier, Claude

doi:10.1007/978-94-009-1732-3_17

Cited by 16 publications

(9 citation statements)

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“…From this perspective, variables can represent many numbers simultaneously; they have no place value and can be selected arbitrarily. In the mathematics education community, we have not had any consistent conceptions for the following pairs of concepts: letters and variables, unknowns and variables, placeholders and variables (Janvier, 1996). Some educators and curriculum developers believe that both words in each of the pairs mean the same thing, but others believe that they represent different concepts.…”

Section: Focusing On the Ideas Of Variablementioning

confidence: 97%

How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable

Nie

Cai

Moyer

2009

ZDM Mathematics Education

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Analyzing the important features of different curricula is critical to understand their effects on students' learning of algebra. Since the concept of variable is fundamental in algebra, this article compares the intended treatments of variable in an NSF-funded standards-based middle school curriculum (CMP) and a more traditionally based curriculum (Glencoe Mathematics). We found that CMP introduces variables as quantities that change or vary, and then it uses them to represent relationships. Glencoe Mathematics, on the other hand, treats variables predominantly as placeholders or unknowns, and then it uses them primarily to represent unknowns in equations. We found strong connections among variables, equation solving, and linear functions in CMP. Glencoe Mathematics, in contrast, emphasizes less on the connections between variables and functions or between algebraic equations and functions, but it does have a strong emphasis on the relation between variables and equation solving.

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Section: Focusing On the Ideas Of Variablementioning

confidence: 97%

How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable

Nie

Cai

Moyer

2009

ZDM Mathematics Education

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“…Do you think it's a good choice? Janvier (1996) points out a particular phenomena related to the modelling approach and student's use of symbols that is widely discussed in the literature (see Clement, Lochhead & Monk, 1981): the frequent misunderstanding of some students in distinguishing between the letters that are the symbolic representation of magnitude units and the ones that stand for variables, especially when students tend to include magnitudes within their algebraic rules: For instance, the same g may represent the constant used in E = mgh, an abbreviated form for gram, or a variable such as a particular gain (that stands for a varying magnitude expressed in dollars). (Janvier, 1996, p. 231) A similar question to the one pointed out by Janvier can be found in our study.…”

Section: Perceiving and Using Letters -A Triple Influencementioning

confidence: 97%

“…We will also discuss the thinking that may have underpinned these choices. Kuchemann (1981) and MacGregor and Stacey (1997) outline some of the ways in which beginning algebra students interpret letters and Janvier (1996) recalls the various interpretations a student can give to letters even when they are firmly established as numbers:…”

Section: Students' Pre-test Ability To Understand and Utilise Algebramentioning

confidence: 98%

Teaching Linear Functions in Context with Graphics Calculators: Students? Responses and the Impact of the Approach on Their Use of Algebraic Symbols

Bardini

Pierce

Stacey

2004

Int J Sci Math Educ

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This study analyses some of the consequences of adopting a functional/ modelling approach to the teaching of algebra. The teaching of one class of 17 students was observed over five weeks, with 15 students undertaking both pre-and post-tests and 6 students and the teacher being interviewed individually. Use of graphics calculators made the predominantly graphical approach feasible. Students made considerable progress in describing linear relationships algebraically. They commented favourably on several aspects of learning concepts through problems in real contexts and were able to set up equations to solve contextualised problems. Three features of the program exerted a 'triple influence' on students' use and understanding of algebraic symbols. Students' concern to express features of the context was evident in some responses, as was the influence of particular contexts selected. Use of graphics calculators affected some students' choice of letters. The functional approach was evident in the meanings ascribed to letters and rules. Students were very positively disposed to the calculators, and interesting differences were observed between the confidence with which they asked questions about the technology and the diffidence with which they asked mathematical questions.

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“…it is in many respects 'generalised arithmetic'" (Booth, 1988, p. 29) and also in Kieran's (1981) proposed sequence for teaching algebraic equations that is "anchored" (p. 322) in arithmetic. Janvier (1996) also argued that algebra involves the use of arithmetic operations on an unknown.…”

Section: Development Of Mathematical Knowledgementioning

confidence: 99%

Sequential development of algebra knowledge: A cognitive analysis

1998

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Learning to operate algebraically is a complex process that is dependent upon extending arithmetic knowledge to the more complex concepts of algebra. Current research has shown a gap between arithmetic and algebraic knowledge and suggests a pre-algebraic level as a step between the two knowledge types. This paper examines arithmetic and algebraic knowledge from a cognitive perspective in an effort to determine what constitutes a pre-algebraic level of understanding. Results of a longitudinal study designed to investigate students' readiness for algebra are presented. Thirty-three students in Grades 7, 8, and 9 participated. A model for the transition from arithmetic to pre-algebra to algebra is proposed and students' understanding of relevant knowledge is discussed.The transition from concrete and numerical foundations of arithmetic to the symbolic and abstract reasoning of algebra has been conceptualised as a didactic cut by Filloy and Rojano (1989) and as a cognitive gap by Herscovics and Linchevski (1994). A consequence of this cut/gap has been recognition of the need for an operational level of pre-algebraic knowledge between arithmetic and algebra. This study proposes a model that depicts the transition from arithmetic to algebra and includes an operational level of pre-algebra. However, instead of explaining this transition as a cut/gap, it is explained in terms of a sequence of development from arithmetic to algebraic knowledge.

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Modeling and the Initiation into Algebra

Cited by 16 publications

References 0 publications

How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable

How a standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable

Teaching Linear Functions in Context with Graphics Calculators: Students? Responses and the Impact of the Approach on Their Use of Algebraic Symbols

Sequential development of algebra knowledge: A cognitive analysis

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