國立虎尾科技大學 |

How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number.

紀錄類型:	書目-語言資料,手稿 : Monograph/item
正題名/作者:	How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number./
作者:	Relaford-Doyle, Josephine.
面頁冊數:	1 online resource (143 pages)
附註:	Source: Dissertations Abstracts International, Volume: 83-10, Section: A.
Contained By:	Dissertations Abstracts International83-10A.
標題:	Mathematics education. -
電子資源:	click for full text (PQDT)
ISBN:	9798209939269

How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number.
Relaford-Doyle, Josephine.

How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number. - 1 online resource (143 pages)

Source: Dissertations Abstracts International, Volume: 83-10, Section: A.

Thesis (Ph.D.)--University of California, San Diego, 2022.

Includes bibliographical references

It is widely assumed within developmental psychology that spontaneously-arising conceptualizations of natural number-those that develop without explicit mathematics instruction-match the formal characterization of natural number given in the Dedekind-Peano Axioms (e.g., Carey, 2004; Leslie et al., 2008; Rips et al., 2008). Specifically, developmental researchers assume that fully-developed conceptualizations of natural number are characterized by knowledge of a starting value 'one' and understanding of the successor principle: that for any natural number n, the next natural number is given by n + 1. This formally-consistent knowledge is thought to emerge spontaneously as children learn how to count, without the need for formal instruction in the axioms of natural number. In developmental psychology, the assumption that spontaneously-arising conceptualizations of natural number match the formal characterization has been taken as unproblematic, and has not been subject to empirical investigation.In this dissertation I seek to provide a more rigorous, thorough, and empirically-grounded characterization of spontaneously-arising natural number concepts. What do we know about natural number, without being explicitly taught? And to what extent are these conceptualizations consistent with the formal mathematical definition given in the Dedekind-Peano Axioms? In the four studies that comprise the dissertation I use an open-ended problem-solving task, a computer-based judgment task, a number-line estimation task, and semi-structured interviews to explore undergraduates' conceptualizations of natural number. I find evidence that undergraduates who have not received explicit instruction in the axioms of natural number often recruit conceptualizations that are at odds with the formal characterization given in the Dedekind-Peano Axioms, and that many students demonstrate formally-consistent reasoning in only the simplest and most foundational mathematical contexts. Taken together, these results call into question the assumption that formally-consistent notions of natural number arise spontaneously from mastery of counting. Based on these findings, I make recommendations for future research into number concept development and discuss practical implications for classroom instruction for concepts that build on the natural number system (for instance, mathematical induction), as well as mathematics teacher development.

Electronic reproduction.
Ann Arbor, Mich. :
ProQuest,
2024

Mode of access: World Wide Web

ISBN: 9798209939269Subjects--Topical Terms:

1148686
Mathematics education.
Subjects--Index Terms:

Natural numberIndex Terms--Genre/Form:

554714
Electronic books.

How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number.
LDR:03725ntm a22003377 4500 001 1149168
005 20241007100303.5
006 m o d
007 cr bn ---uuuuu
008 250605s2022 xx obm 000 0 eng d
020 $a 9798209939269
035 $a (MiAaPQ)AAI28967113
035 $a AAI28967113
040 $a MiAaPQ $b eng $c MiAaPQ $d NTU
100 1 $a Relaford-Doyle, Josephine. $3 1475322
245 1 0 $a How 'Natural' Are the Natural Numbers? Inconsistencies between Formal Axioms and Undergraduates' Conceptualizations of Number.
264 0 $c 2022
300 $a 1 online resource (143 pages)
336 $a text $b txt $2 rdacontent
337 $a computer $b c $2 rdamedia
338 $a online resource $b cr $2 rdacarrier
500 $a Source: Dissertations Abstracts International, Volume: 83-10, Section: A.
500 $a Advisor: Nunez, Rafael.
502 $a Thesis (Ph.D.)--University of California, San Diego, 2022.
504 $a Includes bibliographical references
520 $a It is widely assumed within developmental psychology that spontaneously-arising conceptualizations of natural number-those that develop without explicit mathematics instruction-match the formal characterization of natural number given in the Dedekind-Peano Axioms (e.g., Carey, 2004; Leslie et al., 2008; Rips et al., 2008). Specifically, developmental researchers assume that fully-developed conceptualizations of natural number are characterized by knowledge of a starting value 'one' and understanding of the successor principle: that for any natural number n, the next natural number is given by n + 1. This formally-consistent knowledge is thought to emerge spontaneously as children learn how to count, without the need for formal instruction in the axioms of natural number. In developmental psychology, the assumption that spontaneously-arising conceptualizations of natural number match the formal characterization has been taken as unproblematic, and has not been subject to empirical investigation.In this dissertation I seek to provide a more rigorous, thorough, and empirically-grounded characterization of spontaneously-arising natural number concepts. What do we know about natural number, without being explicitly taught? And to what extent are these conceptualizations consistent with the formal mathematical definition given in the Dedekind-Peano Axioms? In the four studies that comprise the dissertation I use an open-ended problem-solving task, a computer-based judgment task, a number-line estimation task, and semi-structured interviews to explore undergraduates' conceptualizations of natural number. I find evidence that undergraduates who have not received explicit instruction in the axioms of natural number often recruit conceptualizations that are at odds with the formal characterization given in the Dedekind-Peano Axioms, and that many students demonstrate formally-consistent reasoning in only the simplest and most foundational mathematical contexts. Taken together, these results call into question the assumption that formally-consistent notions of natural number arise spontaneously from mastery of counting. Based on these findings, I make recommendations for future research into number concept development and discuss practical implications for classroom instruction for concepts that build on the natural number system (for instance, mathematical induction), as well as mathematics teacher development.
533 $a Electronic reproduction. $b Ann Arbor, Mich. : $c ProQuest, $d 2024
538 $a Mode of access: World Wide Web
650 4 $a Mathematics education. $3 1148686
650 4 $a Higher education. $3 1148448
653 $a Natural number
655 7 $a Electronic books. $2 local $3 554714
690 $a 0280
690 $a 0745
710 2 $a ProQuest Information and Learning Co. $3 1178819
710 2 $a University of California, San Diego. $b Cognitive Science. $3 1186722
773 0 $t Dissertations Abstracts International $g 83-10A.
856 4 0 $u http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=28967113 $z click for full text (PQDT)