The Transition to Formal Thinking in Mathematics
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| Title: | The Transition to Formal Thinking in Mathematics |
|---|---|
| Language: | English |
| Authors: | Tall, David |
| Source: | Mathematics Education Research Journal. Sep 2008 20(2):5-24. |
| Availability: | Mathematics Education Research Group of Australasia. 67 Kokoda Avenue, Wahroonga 2076, Australia. Fax: +61-9688-5001; e-mail: vp.conferences@merga.net.au; Web Site: http://www.merga.net.au/publications/merj.php |
| Peer Reviewed: | Y |
| Physical Description: | |
| Page Count: | 20 |
| Publication Date: | 2008 |
| Document Type: | Journal Articles Reports - Descriptive |
| Education Level: | Higher Education Postsecondary Education |
| Descriptors: | Mathematical Logic, Mathematics Instruction, Mathematical Concepts, College Mathematics, Computation, Mathematics Education, Secondary School Mathematics, Cognitive Development, Cognitive Processes, Genetics, Concept Formation, Formal Operations, Learning Theories, Mathematics |
| ISSN: | 1033-2170 |
| Abstract: | This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of "three worlds of mathematics"--the "conceptual-embodied" world based on perception, action and thought experiment, the "proceptual-symbolic" world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the "axiomatic-formal" world of set-theoretic concept definitions and mathematical proof. Each "world" has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds. (Contains 8 figures.) |
| Abstractor: | As Provided |
| Number of References: | 26 |
| Entry Date: | 2008 |
| Accession Number: | EJ820231 |
| Database: | ERIC |
| FullText | Text: Availability: 0 CustomLinks: – Url: https://eric.ed.gov/contentdelivery/servlet/ERICServlet?accno=EJ820231 Name: ERIC Full Text Category: fullText Text: Full Text from ERIC |
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| Header | DbId: eric DbLabel: ERIC An: EJ820231 AccessLevel: 3 PubType: Academic Journal PubTypeId: academicJournal PreciseRelevancyScore: 0 |
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| Items | – Name: Title Label: Title Group: Ti Data: The Transition to Formal Thinking in Mathematics – Name: Language Label: Language Group: Lang Data: English – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Tall%2C+David%22">Tall, David</searchLink> – Name: TitleSource Label: Source Group: Src Data: <searchLink fieldCode="SO" term="%22Mathematics+Education+Research+Journal%22"><i>Mathematics Education Research Journal</i></searchLink>. Sep 2008 20(2):5-24. – Name: Avail Label: Availability Group: Avail Data: Mathematics Education Research Group of Australasia. 67 Kokoda Avenue, Wahroonga 2076, Australia. Fax: +61-9688-5001; e-mail: vp.conferences@merga.net.au; Web Site: http://www.merga.net.au/publications/merj.php – Name: PeerReviewed Label: Peer Reviewed Group: SrcInfo Data: Y – Name: PhysDesc Label: Physical Description Group: PhysDesc Data: PDF – Name: Pages Label: Page Count Group: Src Data: 20 – Name: DatePubCY Label: Publication Date Group: Date Data: 2008 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Journal Articles<br />Reports - Descriptive – Name: Audience Label: Education Level Group: Audnce Data: <searchLink fieldCode="EL" term="%22Higher+Education%22">Higher Education</searchLink><br /><searchLink fieldCode="EL" term="%22Postsecondary+Education%22">Postsecondary Education</searchLink> – Name: Subject Label: Descriptors Group: Su Data: <searchLink fieldCode="DE" term="%22Mathematical+Logic%22">Mathematical Logic</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Instruction%22">Mathematics Instruction</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematical+Concepts%22">Mathematical Concepts</searchLink><br /><searchLink fieldCode="DE" term="%22College+Mathematics%22">College Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Computation%22">Computation</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+Education%22">Mathematics Education</searchLink><br /><searchLink fieldCode="DE" term="%22Secondary+School+Mathematics%22">Secondary School Mathematics</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Development%22">Cognitive Development</searchLink><br /><searchLink fieldCode="DE" term="%22Cognitive+Processes%22">Cognitive Processes</searchLink><br /><searchLink fieldCode="DE" term="%22Genetics%22">Genetics</searchLink><br /><searchLink fieldCode="DE" term="%22Concept+Formation%22">Concept Formation</searchLink><br /><searchLink fieldCode="DE" term="%22Formal+Operations%22">Formal Operations</searchLink><br /><searchLink fieldCode="DE" term="%22Learning+Theories%22">Learning Theories</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics%22">Mathematics</searchLink> – Name: ISSN Label: ISSN Group: ISSN Data: 1033-2170 – Name: Abstract Label: Abstract Group: Ab Data: This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of "three worlds of mathematics"--the "conceptual-embodied" world based on perception, action and thought experiment, the "proceptual-symbolic" world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the "axiomatic-formal" world of set-theoretic concept definitions and mathematical proof. Each "world" has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds. (Contains 8 figures.) – Name: AbstractInfo Label: Abstractor Group: Ab Data: As Provided – Name: Ref Label: Number of References Group: RefInfo Data: 26 – Name: DateEntry Label: Entry Date Group: Date Data: 2008 – Name: AN Label: Accession Number Group: ID Data: EJ820231 |
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| RecordInfo | BibRecord: BibEntity: Languages: – Text: English PhysicalDescription: Pagination: PageCount: 20 StartPage: 5 Subjects: – SubjectFull: Mathematical Logic Type: general – SubjectFull: Mathematics Instruction Type: general – SubjectFull: Mathematical Concepts Type: general – SubjectFull: College Mathematics Type: general – SubjectFull: Computation Type: general – SubjectFull: Mathematics Education Type: general – SubjectFull: Secondary School Mathematics Type: general – SubjectFull: Cognitive Development Type: general – SubjectFull: Cognitive Processes Type: general – SubjectFull: Genetics Type: general – SubjectFull: Concept Formation Type: general – SubjectFull: Formal Operations Type: general – SubjectFull: Learning Theories Type: general – SubjectFull: Mathematics Type: general Titles: – TitleFull: The Transition to Formal Thinking in Mathematics Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Tall, David IsPartOfRelationships: – BibEntity: Dates: – D: 01 M: 09 Type: published Y: 2008 Identifiers: – Type: issn-print Value: 1033-2170 Numbering: – Type: volume Value: 20 – Type: issue Value: 2 Titles: – TitleFull: Mathematics Education Research Journal Type: main |
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