### Citations

332 |
The child's understanding of number
- Gelman, Gallistel
- 1986
(Show Context)
Citation Context ... and entrench a rich concept of counting numbers, positive integers as represented by the integer list and base 10 notation, and as participate in operations of addition and subtraction (Fuson, 1988; =-=Gelman & Gallistel, 1978-=-). The representation of the positive integers is built by the child through mastering the counting algorithm, which in turn implements the successor relation among integers. For each positive integer... |

183 |
Insides and essence: Early understandings of the non-obvious.
- Gelman, Wellman
- 1991
(Show Context)
Citation Context ...nger than the typical dog. Whatever essential characteristics of dogs the child represents (e.g., that they are born from dog parents or that it is something about their insides that makes them dogs; =-=Gelman & Wellman, 1991-=-; Keil, 1989) are true of dogs before and after this change. In other words, children diVerentiate the concept dog into breeds without changing their initial concept, either in its essential features ... |

174 |
Children’s counting and concepts of number,
- Fuson
- 1988
(Show Context)
Citation Context ...ldren create, and entrench a rich concept of counting numbers, positive integers as represented by the integer list and base 10 notation, and as participate in operations of addition and subtraction (=-=Fuson, 1988-=-; Gelman & Gallistel, 1978). The representation of the positive integers is built by the child through mastering the counting algorithm, which in turn implements the successor relation among integers.... |

108 | How do scientists think? Capturing the dynamics of conceptual change in science, in Cognitive Models of Science, edited by R.N. - Nersessian - 1992 |

106 |
The Essential Tension. Selected Studies in Scientific Tradition
- Kuhn
- 1977
(Show Context)
Citation Context ...bed “bootstrapping processes” and have been sketched by many historians and philosophers of science, as well as cognitive scientists (e.g., Carey, 2004; Gentner et al., 1997; Hartnett & Gelman, 1998; =-=Kuhn, 1977-=-; Nersessian, 1992). At its heart, bootstrapping makes use of various modeling techniques—creating analogies between diVerent domains, limiting case analyses, thought experiments, and so on. It also m... |

102 |
Developing conceptual understanding and procedural skill in mathematics: An iterative process.
- Rittle-Johnson, Siegler, et al.
- 2001
(Show Context)
Citation Context ...n ordering two decimals such as 2.09 and 2.9 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; Gelman, 1991; Moss & Case, 1999), placing a number like .685 on a number line that goes from 0 to 1 (=-=Rittle-Johnson, Siegler, & Alibali, 2001-=-), and lining up decimals such as 5.1 and .46 so as to add or subtract them (Hiebert & Wearne, 1986). Such persistent problems with understanding fraction and decimal notation, including seemingly sim... |

80 |
The child's conception of geometry.
- Piaget, Inhelder, et al.
- 1960
(Show Context)
Citation Context ...children Wrst understand geometric entities such as lines and circles as concrete physical marks on paper, and deny that a line could connect two points in empty space or could continue oV the paper (=-=Piaget, Inhelder, & Szeminska, 1964-=-). The capacity for geometric abstraction to support these latter thought experiments may be important for the construction of a continuous model of physical extent. Third, in our view, the most impor... |

77 |
Knowledge acquisition: Enrichment or conceptual change? In
- Carey
- 1991
(Show Context)
Citation Context ...t the inWnite divisibility of material objects (Fischbein, Tirosh, Stavy, & Oster, 1990; Smith et al., 1997; Stavy & Tirosh, 2000), the amount of space occupied by material objects, and their weight (=-=Carey, 1991-=-; Smith, 2005; Smith et al., 1997). The latter work has conWrmed that many students can at best imagine only a limited number of divisions before the matter disappears and the amount of weight or occu... |

62 |
Developing children's understanding of the rational numbers: A new model and an experimental curriculum.
- Moss, Case
- 1999
(Show Context)
Citation Context ...her & Peled, 1986, for Israel). Similarly, researchers have found persistent diYculty in ordering two decimals such as 2.09 and 2.9 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; Gelman, 1991; =-=Moss & Case, 1999-=-), placing a number like .685 on a number line that goes from 0 to 1 (Rittle-Johnson, Siegler, & Alibali, 2001), and lining up decimals such as 5.1 and .46 so as to add or subtract them (Hiebert & Wea... |

49 | On differentiation: A case study of the development of the concepts of size, weight, and density.
- Smith, Carey, et al.
- 1985
(Show Context)
Citation Context ...successful at diagnosing key elements of children’s commonsense theories of matter—namely that amount, spatial extent, and weight of matter are continuous, extensive physical quantities (Carey, 1991; =-=Smith, Carey, & Wiser, 1985-=-; Smith et al., 1997). Students were asked a progressive series of questions that led them from thinking about macroscopic pieces of Styrofoam they held in their hands to thought experiments about pie... |

44 |
Junior high school pupils understanding of the particulate nature of matter: An interview study
- Novick, Nussbaum
- 1978
(Show Context)
Citation Context ...the major diYculties students have in initially learning about atoms is accepting the basic tenet that there can be empty space between atoms (Lee, Eichinger, Anderson, Berkheimer, & Blakeslee, 1993; =-=Novick & Nussbaum, 1978-=-; Pfundt, 1981). However, children do not initially consider matter continuous in this sense, nor do they initially conceive of weight or diVerent aspects of spatial extent (e.g., length, area, or vol... |

38 |
Changing middle school students‟ conceptions of matter and molecules.
- Lee, Eichinger, et al.
- 1993
(Show Context)
Citation Context ...ong is their intuition that matter is continuous, one of the major diYculties students have in initially learning about atoms is accepting the basic tenet that there can be empty space between atoms (=-=Lee, Eichinger, Anderson, Berkheimer, & Blakeslee, 1993-=-; Novick & Nussbaum, 1978; Pfundt, 1981). However, children do not initially consider matter continuous in this sense, nor do they initially conceive of weight or diVerent aspects of spatial extent (e... |

34 |
Splitting, similarity and rate of change: A new approach to multiplication and exponential functions.
- Confrey
- 1994
(Show Context)
Citation Context ...erations such as splitting, sharing, folding, comparing, and perceiving proportionality, that draw on a qualitative appreciation of some aspects of the inferential role of rational number and ratios (=-=Confrey, 1994-=-; Moss & Case, 1999; Resnick & Singer, 1993). Indeed, Mix, Levine, and Huttenlocher (1999) have shown that even preschool children can manipulate models of physical quantities based on parts and ratio... |

33 |
Epigenetic foundations of knowledge structures: Initial and transcendent constructions. In
- Gelman
- 1991
(Show Context)
Citation Context ...gland; and Nesher & Peled, 1986, for Israel). Similarly, researchers have found persistent diYculty in ordering two decimals such as 2.09 and 2.9 (Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981; =-=Gelman, 1991-=-; Moss & Case, 1999), placing a number like .685 on a number line that goes from 0 to 1 (Rittle-Johnson, Siegler, & Alibali, 2001), and lining up decimals such as 5.1 and .46 so as to add or subtract ... |

33 |
Early understanding of number: Paths or barriers to the construction of new understandings?
- Hartnett, Gelman
- 1998
(Show Context)
Citation Context ... culturally constructed vastly earlier than rational numbers. Second, teaching students about fractions and decimals is notoriously diYcult (e.g., Bright, Behr, Post, & Wachsmuth, 1988; Gelman, 1991; =-=Hartnett & Gelman, 1998-=-). There is strong resistance to change; even after instruction, the errors children make in tasks designed to reveal their understanding of fractions are typically whole number intrusions. For exampl... |

33 |
Procedures over concepts: The acquisition of decimal number knowledge. In
- Hiebert, Wearne
- 1986
(Show Context)
Citation Context ... & Case, 1999), placing a number like .685 on a number line that goes from 0 to 1 (Rittle-Johnson, Siegler, & Alibali, 2001), and lining up decimals such as 5.1 and .46 so as to add or subtract them (=-=Hiebert & Wearne, 1986-=-). Such persistent problems with understanding fraction and decimal notation, including seemingly simple operations over fractions and decimals, may actually reXect deep conceptual diYculties in under... |

31 |
1). Bootstrapping and the origin of concepts. Daedalus
- Carey
- 2004
(Show Context)
Citation Context ...ional power in the course of conceptual change have been dubbed “bootstrapping processes” and have been sketched by many historians and philosophers of science, as well as cognitive scientists (e.g., =-=Carey, 2004-=-; Gentner et al., 1997; Hartnett & Gelman, 1998; Kuhn, 1977; Nersessian, 1992). At its heart, bootstrapping makes use of various modeling techniques—creating analogies between diVerent domains, limiti... |

31 | Early fraction calculation ability.
- Mix, Levine, et al.
- 1999
(Show Context)
Citation Context ...ed the conceptual change position on the grounds that protonumerical understanding of ratios and division is part of children’s mental models for reasoning quantitively about objects and space (e.g., =-=Mix et al., 1999-=-). While this is so, and is presupposed by bootstrapping accounts that draw on these representations in the construction of representations of rational number, these are models of quantities in genera... |

28 |
Commensurability, comparability, communicability
- Kuhn
- 1983
(Show Context)
Citation Context ...requires conceptual change requires distinguishing conceptual change from knowledge enrichment, a notoriously diYcult but not impossible task (see Carey, 1991; Hartnett & Gelman, 1998; Kitcher, 1988; =-=Kuhn, 1982-=-). Conceptual changes involve diVerentiations and coalescences, such that the extension of a concept and its relations to other concepts are qualitatively diVerent after the change than before it. The... |

26 |
Order and equivalence of rational numbers: A clinical teaching experiment.
- Behr, Wachsmuth, et al.
- 1984
(Show Context)
Citation Context ...ering tasks, such as determining whether 1/56 is larger than 1/75. Many researchers in diVerent countries have found that this diYculty persists for some children through the high school years (e.g., =-=Behr, Wachsmuth, Post, & Lesh, 1984-=-, for the US; Kerslake, 1986, for England; and Nesher & Peled, 1986, for Israel). Similarly, researchers have found persistent diYculty in ordering two decimals such as 2.09 and 2.9 (Carpenter, Corbit... |

26 |
Teaching for understanding: A study of students’ preinstruction theories of matter and a comparison of the effectiveness of two approaches to teaching about matter and density
- Smith, Maclin, et al.
- 1997
(Show Context)
Citation Context ... Raz, 2003). Further, when engaged in thought experiments about the repeated halving of matter, many grade 8–12 students claim that the process is endless: there is always some matter left to divide (=-=Smith, Maclin, Grosslight, & Davis, 1997-=-; Stavy & Tirosh, 2000). Similarly when engaged in thought experiments about the serial dilution of sugar or salt water solutions, students often claim that there will always be some sugar or salt lef... |

25 |
Protoquantitative origins of ratio reasoning
- Resnick, Singer
- 1993
(Show Context)
Citation Context ...g, folding, comparing, and perceiving proportionality, that draw on a qualitative appreciation of some aspects of the inferential role of rational number and ratios (Confrey, 1994; Moss & Case, 1999; =-=Resnick & Singer, 1993-=-). Indeed, Mix, Levine, and Huttenlocher (1999) have shown that even preschool children can manipulate models of physical quantities based on parts and ratios (e.g., know that 1/2 a circle added to 1/... |

16 |
Linking phenomena with competing underlying models: A software tool for introducing students to the particulate model of matter.
- Snir, Smith, et al.
- 2003
(Show Context)
Citation Context ... learning about atoms, grades 5–7 students draw continuous models of liquids and are puzzled to Wnd that the volume of an alcohol and water mixture was less than the sum of the volume of each liquid (=-=Snir, Smith, & Raz, 2003-=-). Further, when engaged in thought experiments about the repeated halving of matter, many grade 8–12 students claim that the process is endless: there is always some matter left to divide (Smith, Mac... |

11 |
Identifying fractions on a number line
- Bright, Behr, et al.
- 1988
(Show Context)
Citation Context ...ve, as well as the historical fact that integers were culturally constructed vastly earlier than rational numbers. Second, teaching students about fractions and decimals is notoriously diYcult (e.g., =-=Bright, Behr, Post, & Wachsmuth, 1988-=-; Gelman, 1991; Hartnett & Gelman, 1998). There is strong resistance to change; even after instruction, the errors children make in tasks designed to reveal their understanding of fractions are typica... |

11 |
Analogical Reasoning and Conceptual Change: A Case Study of Johannes Kepler
- Gentzer, Brem, et al.
- 1997
(Show Context)
Citation Context ...n the course of conceptual change have been dubbed “bootstrapping processes” and have been sketched by many historians and philosophers of science, as well as cognitive scientists (e.g., Carey, 2004; =-=Gentner et al., 1997-=-; Hartnett & Gelman, 1998; Kuhn, 1977; Nersessian, 1992). At its heart, bootstrapping makes use of various modeling techniques—creating analogies between diVerent domains, limiting case analyses, thou... |

11 |
Bootstrapping processes in the development of students' commonsense matter theories: Using analogical mappings, thought experiments, and learning to measure to promote conceptual restructuring
- Smith
- 2007
(Show Context)
Citation Context ... divisibility of material objects (Fischbein, Tirosh, Stavy, & Oster, 1990; Smith et al., 1997; Stavy & Tirosh, 2000), the amount of space occupied by material objects, and their weight (Carey, 1991; =-=Smith, 2005-=-; Smith et al., 1997). The latter work has conWrmed that many students can at best imagine only a limited number of divisions before the matter disappears and the amount of weight or occupied space go... |

10 |
The child as parent of the scientist
- Kitcher
- 1988
(Show Context)
Citation Context ...ational number requires conceptual change requires distinguishing conceptual change from knowledge enrichment, a notoriously diYcult but not impossible task (see Carey, 1991; Hartnett & Gelman, 1998; =-=Kitcher, 1988-=-; Kuhn, 1982). Conceptual changes involve diVerentiations and coalescences, such that the extension of a concept and its relations to other concepts are qualitatively diVerent after the change than be... |

8 |
Fractions: Children’s strategies and errors: A report of the strategies and error in secondary mathematics project
- Kerslake
- 1986
(Show Context)
Citation Context ...larger than 1/75. Many researchers in diVerent countries have found that this diYculty persists for some children through the high school years (e.g., Behr, Wachsmuth, Post, & Lesh, 1984, for the US; =-=Kerslake, 1986-=-, for England; and Nesher & Peled, 1986, for Israel). Similarly, researchers have found persistent diYculty in ordering two decimals such as 2.09 and 2.9 (Carpenter, Corbitt, Kepner, Lindquist, & Reys... |

3 |
990), The Autonomy of Mental Models
- FISCHBEIN, TIROSH, et al.
- 1990
(Show Context)
Citation Context ... Psychology 51 (2005) 101–140 107 focus of Piaget and Inhelder’s work 2 (Fischbein et al., 1979). Later work has also examined children’s reasoning about the inWnite divisibility of material objects (=-=Fischbein, Tirosh, Stavy, & Oster, 1990-=-; Smith et al., 1997; Stavy & Tirosh, 2000), the amount of space occupied by material objects, and their weight (Carey, 1991; Smith, 2005; Smith et al., 1997). The latter work has conWrmed that many s... |

2 |
Decimals: Results and implications from the second NAEP mathematics assessment
- Carpenter, Corbitt, et al.
- 1981
(Show Context)
Citation Context ...ost, & Lesh, 1984, for the US; Kerslake, 1986, for England; and Nesher & Peled, 1986, for Israel). Similarly, researchers have found persistent diYculty in ordering two decimals such as 2.09 and 2.9 (=-=Carpenter, Corbitt, Kepner, Lindquist, & Reys, 1981-=-; Gelman, 1991; Moss & Case, 1999), placing a number like .685 on a number line that goes from 0 to 1 (Rittle-Johnson, Siegler, & Alibali, 2001), and lining up decimals such as 5.1 and .46 so as to ad... |

1 |
Understanding zero and inWnity in the early school years. Unpublished doctoral dissertation
- Evans
- 1983
(Show Context)
Citation Context ...e 1 and 2 and the majority of grade 3 and 4 students can be led to induce and articulate the principle that there is no biggest integer and that counting numbers can always be extended by adding one (=-=Evans, 1983-=-; Falk, Gassner, Ben-Zoor, & BenSimon, 1986; Hartnett & Gelman, 1998). To date, however, there has only been one study that probes elementary school children’s understanding of the inWnite divisibilit... |

1 |
How do children cope with the inWnity of numbers
- Falk, Gassner, et al.
- 1986
(Show Context)
Citation Context ...r, Ben-Zoor, & BenSimon, 1986; Hartnett & Gelman, 1998). To date, however, there has only been one study that probes elementary school children’s understanding of the inWnite divisibility of numbers (=-=Falk et al., 1986-=-). These researchers engaged elementary school students in playing repeated rounds of a two-person game where the winner was the person who picked the smaller positive rational number. Children were a... |

1 |
The intuition of inWnity
- Fischbein, Tirosh, et al.
- 1979
(Show Context)
Citation Context ...esearch has suggested that children’s earliest understanding of inWnity is as a property of processes (being endless), rather than as an amount (or number-like object) that has an order of magnitude (=-=Fischbein, Tirosh, & Hess, 1979-=-; Monaghan, 2001). There is converging evidence from diVerent in depth clinical interviews that many grade 1 and 2 and the majority of grade 3 and 4 students can be led to induce and articulate the pr... |

1 |
To know mathematics is to go beyond thinking that “fractions aren’t numbers
- Gelman, Cohen, et al.
- 1989
(Show Context)
Citation Context ...g of fractions are typically whole number intrusions. For example, they say that 1/56 is smaller than 1/75 because 56 is smaller than 75, that 2.9 is smaller than 2.09 because 29 is smaller than 209 (=-=Gelman, Cohen, & Hartnett, 1989-=-; Moss & Case, 1999). Third, various reXections of conceptual understanding of fractions—most notably awareness of the existence of numbers between integers, understanding the relation between numerat... |

1 | et al. / Cognitive Psychology 51 (2005) 101–140 - Smith - 1989 |

1 |
Young peoples’ ideas of inWnity
- Monaghan
- 2001
(Show Context)
Citation Context ...en’s earliest understanding of inWnity is as a property of processes (being endless), rather than as an amount (or number-like object) that has an order of magnitude (Fischbein, Tirosh, & Hess, 1979; =-=Monaghan, 2001-=-). There is converging evidence from diVerent in depth clinical interviews that many grade 1 and 2 and the majority of grade 3 and 4 students can be led to induce and articulate the principle that the... |

1 | The child’s construction of space - Piaget, Inhelder - 1956 |

1 |
The atom—the Wnal link in the division process or the Wrst building block?. Chimica didactica
- Pfundt
- 1981
(Show Context)
Citation Context ...ents have in initially learning about atoms is accepting the basic tenet that there can be empty space between atoms (Lee, Eichinger, Anderson, Berkheimer, & Blakeslee, 1993; Novick & Nussbaum, 1978; =-=Pfundt, 1981-=-). However, children do not initially consider matter continuous in this sense, nor do they initially conceive of weight or diVerent aspects of spatial extent (e.g., length, area, or volume) as contin... |

1 |
Using conceptual models to teach inner city students about density: The promise and the prerequisites. Final report submitted to the McDonnell Foundation
- Smith, Grosslight, et al.
- 1994
(Show Context)
Citation Context ...tter itself, matter’s spatial extent, or its weight, and that understanding the continuity of matter’s spatial extent reliably precedes the understanding of the continuity of its weight (Carey, 1991; =-=Smith, Grosslight, Davis, Unger, & Snir, 1994-=-). It goes beyond the earlier studies by including parallel thought experiments about all three quantities and more carefully probing the thought experiment about matter (i.e., making it clearer wheth... |

1 |
The nature of the intuitive rule ‘Everything can be divided
- Stavy, Tirosh
- 2000
(Show Context)
Citation Context ...ght experiments about the repeated halving of matter, many grade 8–12 students claim that the process is endless: there is always some matter left to divide (Smith, Maclin, Grosslight, & Davis, 1997; =-=Stavy & Tirosh, 2000-=-). Similarly when engaged in thought experiments about the serial dilution of sugar or salt water solutions, students often claim that there will always be some sugar or salt left in the solution (Tir... |