The learning and teaching of school algebra.

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The learning and teaching of school algebra.

The learning and teaching of school algebra.

01 Jan 1992-pp 390-419

About: The article was published on 1992-01-01 and is currently open access. It has received 736 citations till now. The article focuses on the topics: Teaching and learning center & Experiential learning.

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Citations

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Journal Article•DOI•

Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations.

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Bethany Rittle-Johnson¹, Jon R. Star²•Institutions (2)

Vanderbilt University¹, Harvard University²

01 Aug 2007-Journal of Educational Psychology

TL;DR: In this paper, the authors randomly assigned 70 seventh grade students to learn about algebra equation solving by either comparing and contrasting alternative solution methods or reflecting on the same solution methods one at a time, and found that students in the compare group had greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge.

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Abstract: Encouraging students to share and compare solution methods is a key component of reform efforts in mathematics, and comparison is emerging as a fundamental learning mechanism. To experimentally evaluate the effects of comparison for mathematics learning, the authors randomly assigned 70 seventhgrade students to learn about algebra equation solving by either (a) comparing and contrasting alternative solution methods or (b) reflecting on the same solution methods one at a time. At posttest, students in the compare group had made greater gains in procedural knowledge and flexibility and comparable gains in conceptual knowledge. These findings suggest potential mechanisms behind the benefits of comparing contrasting solutions and ways to support effective comparison in the classroom.

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473 citations

Cites background from "The learning and teaching of school..."

...Historically, algebra has represented students’ first sustained exposure to the abstraction and symbolism that makes mathematics powerful (Kieran, 1992)....
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...Current mathematics curricula typically do not focus sufficiently on flexible and meaningful solving of equations (Kieran, 1992)....
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Book Chapter•DOI•

The gains and the pitfalls of reification — The case of algebra

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Anna Sfard¹, Liora Linchevski¹•Institutions (1)

Hebrew University of Jerusalem¹

01 Mar 1994-Educational Studies in Mathematics

TL;DR: In this article, the main focus is on the versatility and adaptability of student's algebraic knowledge and the importance of what one actually sees in algebraic symbols depending on the requirements of the problem to which they are applied.

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Abstract: Algebraic symbols do not speak for themselves. What one actually sees in them depends on the requirements of the problem to which they are applied. Not less important, it depends on what one is able to perceive and prepared to notice. It is this last statement which becomes the leading theme of this article. The main focus is on the versatility and adaptability of student’s algebraic knowledge

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446 citations

Component•DOI•

Does Understanding the Equal Sign Matter? Evidence from Solving Equations

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Eric J. Knuth, Ana C. Stephens, Nicole M. McNeil, Martha W. Alibali

01 Jul 2006-Journal for Research in Mathematics Education

TL;DR: In this paper, the authors focus on middle school students' understanding of the equal sign and its relation to performance solving algebraic equations and find that many students lack a sophisticated understanding of this concept and that their understanding is associated with performance on equation-solving items.

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Abstract: Given its important role in mathematics as well as its role as a gatekeeper to future educational and employment opportunities, algebra has become a focal point of both reform and research efforts in mathematics education. Understanding and using algebra is dependent on understanding a number of fundamental concepts, one of which is the concept of equality. This article focuses on middle school students’ understanding of the equal sign and its relation to performance solving algebraic equations. The data indicate that many students lack a sophisticated understanding of the equal sign and that their understanding of the equal sign is associated with performance on equation-solving items. Moreover, the latter finding holds even when controlling for mathematics ability (as measured by standardized achievement test scores). Implications for instruction and curricular design are discussed.

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406 citations

Cites background from "The learning and teaching of school..."

...Kieran (1992) suggested that "one of the requirements for generating and adequately interpreting structural epresentations such as equations i a conception of the symmetric This content downloaded from 128.192.114.19 on Mon, 25 Aug 2014 22:22:47 UTC All use subject to JSTOR Terms and…...
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..., Lacampagne, Blair, & Kaput, 1995; National Council of Teachers of Mathematics [NCTM], 2000) and research (e.g., Bednarz, Kieran, & Lee, 1996; Kaput, Carraher, & Blanton, in press; Kieran, 1992; Olive, Izsak, & Blanton, 2002; RAND Mathematics Study Panel, 2003; Wagner & Kieran, 1989) in mathematics education....
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...…Blair, & Kaput, 1995; National Council of Teachers of Mathematics [NCTM], 2000) and research (e.g., Bednarz, Kieran, & Lee, 1996; Kaput, Carraher, & Blanton, in press; Kieran, 1992; Olive, Izsak, & Blanton, 2002; RAND Mathematics Study Panel, 2003; Wagner & Kieran, 1989) in mathematics education....
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Journal Article•DOI•

The real story behind story problems: Effects of representations on quantitative reasoning.

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Kenneth R. Koedinger¹, Mitchell J. Nathan¹•Institutions (1)

Carnegie Mellon University¹

01 Apr 2004-The Journal of the Learning Sciences

TL;DR: The authors explored how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning, finding that students were more successful solving simple algebra story problems than solving mathematically equivalent equations.

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Abstract: This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than solving mathematically equivalent equations. Contrary to some views of situated cognition, this result is not simply a consequence of situated world knowledge facilitating problem-solving performance, but rather a consequence of student difficulties with comprehending the formal symbolic representation of quantitative relations. We draw on analyses of students' strategies and errors as the basis for a cognitive process explanation of when, why, and how differences in problem representation affect problem solving. We conclude that differences in external representations can affect performance and learning when one representation is easier to comprehend than anoth...

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391 citations

Cites background from "The learning and teaching of school..."

...problem research (Reed, 1998), and algebra research (Bednarz, Kieran, & Lee, 1996; Kieran, 1992) do...
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...…learning research (Kilpatrick, Swafford, & Findell, 2001), story problem research (Reed, 1998), and algebra research (Bednarz, Kieran, & Lee, 1996; Kieran, 1992) do not reference any such studies. and last columns in Table 1), we added an intermediate problem representation we refer to as word…...
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...…classes of algebra problems more difficult than the ones used, for instance, problems where the unknown is referenced more than once and possibly on both sides of the equal sign, such as 5.8x – 25 = 5.5x (e.g., Bednarz & Janvier, 1996; Kieran, 1992; Koedinger, Alibali, & Nathan, submitted)....
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...However, we found many students using a variety of informal strategies as well (cf. Hall et al., 1989; Katz, Friedman, Bennett, & Berger, 1996; Kieran, 1992; Resnick, 1987; Tabachneck, Koedinger, & Nathan, 1994)....
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Journal Article•DOI•

How Students Understand Physics Equations

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Bruce L. Sherin

01 Dec 2001-Cognition and Instruction

TL;DR: In this article, the authors argue that successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work, and they learn to associate a simple conceptual schema with a pattern of symbols in an equation.

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Abstract: What does it mean to understand a physics equation? The use of formal expressions in physics is not just a matter of the rigorous and routinized application of principles, followed by the formal manipulation of expressions to obtain an answer. Rather, successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work. More specifically, students learn to understand physics equations in terms of a vocabulary of elements that I call symbolic forms. Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. This hypothesis has implications for how we should understand what must be taught and learned in physics classrooms. From the point of view of improving instruction, it is absolutely critical to acknowledge that physics expertise involves this more flexible and generative understanding of equations, and our instruction should be geared toward helping students to acquire this understanding. ...

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376 citations

Cites background from "The learning and teaching of school..."

...Some researchers have contrasted two broad stances that students adopt toward equations: the process stance and the object stance (Herscovics & Kieran, 1980; Kieran, 1992; Sfard, 1987, 1991)....
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...Kieran (1992) argued that learning to adopt the object stance is a develop- mental achievement, with the process stance more natural early in instruction....
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...In addition, researchers have contrasted two broad stances that students are said to adopt toward equations: the process and object stances (Herscovics & Kieran, 1980; Kieran, 1992; Sfard, 1987, 1991)....
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References

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Open Access

More filters

Teachers' thought processes

[...]

C. M. Clark

01 Jan 1986

1,897 citations

Journal Article•DOI•

On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin

[...]

Anna Sfard¹•Institutions (1)

Hebrew University of Jerusalem¹

01 Feb 1991-Educational Studies in Mathematics

TL;DR: In this article, a theoretical framework for investigating the role of algorithms in mathematical thinking is presented, and it is shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.

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Abstract: This paper presents a theoretical framework for investigating the role of algorithms in mathematical thinking. In the study, a combined ontological-psychological outlook is applied. An analysis of different mathematical definitions and representations brings us to the conclusion that abstract notions, such as number or function, can be conceived in two fundamentally different ways: structurally-as objects, and operationally-as processes. These two approaches, although ostensibly incompatible, are in fact complementary. It will be shown that the processes of learning and of problem-solving consist in an intricate interplay between operational and structural conceptions of the same notions.

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1,804 citations

Book•

Mathematical Thought from Ancient to Modern Times

[...]

Morris Kline

01 Jan 1972

TL;DR: In this article, a comprehensive history of mathematical ideas and the careers of the men responsible for them is presented, focusing on geometry and trigonometry, and the role of mathematics in the medieval and early modern periods.

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Abstract: This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the nineteenth century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Goedel on recent mathematical study.

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1,382 citations

Book•

Mathematics as an Educational Task

[...]

Hans Freudenthal

31 Dec 1972

TL;DR: In this article, the authors discuss the development of the number concept from intuition to algorithmizing and rationalizing, from the Algebraic Principle to the Global Organization of Algebra.

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Abstract: I. The Mathematical Tradition.- II. Mathematics Today.- III. Tradition and Education.- IV. Use and Aim of Mathematics Instruction.- V. The Socratic Method.- VI. Re-invention.- VII. Organization of a Field by Mathematizing.- VIII. Mathematical Rigour.- IX. Instruction.- X. The Mathematics Teacher.- XI. The Number Concept - Objective Accesses.- XII. Developing the Number Concept from Intuitive Methods to Algorithmizing and Rationalizing.- XIII. Development of the Number Concept - The Algebraic Method.- XIV. Development of the Number Concept - From the Algebraic Principle to the Global Organization of Algebra.- XV. Sets and Functions.- XVI. The Case of Geometry.- XVII. Analysis.- XVIII. Probability and Statistics.- XIX. Logic.- Appendix I. Piaget and the Piaget School's Investigations on the Development of Mathematical Notions.- Appendix II. Papers of the Author on Mathematical Instruction.

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1,137 citations

Journal Article•DOI•

Functions, Graphs, and Graphing: Tasks, Learning, and Teaching

[...]

Gaea Leinhardt, Orit Zaslavsky, Mary Kay Stein¹•Institutions (1)

University of Pittsburgh¹

01 Jan 1990-Review of Educational Research

TL;DR: This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical.

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Abstract: This review of the introductory instructional substance of functions and graphs analyzes research on the interpretation and construction tasks associated with functions and some of their representations: algebraic, tabular, and graphical. The review also analyzes the nature of learning in terms of intuitions and misconceptions, and the plausible approaches to teaching through sequences, explanations, and examples. The topic is significant because of (a) the increased recognition of the organizing power of the concept of functions from middle school mathematics through more advanced topics in high school and college, and (b) the symbolic connections that represent potentials for increased understanding between graphical and algebraic worlds. This is a review of a specific part of the mathematics subject mailer and how it is learned and may be taught; this specificity reflects the issues raised by recent theoretical research concerning how specific context and content contribute to learning and meaning.

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870 citations