Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment View Full Text


Ontology type: schema:ScholarlyArticle     


Article Info

DATE

1999-03

AUTHORS

John Mason, Mary Spence

ABSTRACT

Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam: Myself when young did eagerly frequent, Doctor and Saint and heard great argument, About it and about: but ever more Came out by the same door as in I went (p. 341). Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions ‘come to mind’. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform. More... »

PAGES

135-161

References to SciGraph publications

  • 1994-03. Growth in mathematical understanding: How can we characterise it and how can we represent it? in EDUCATIONAL STUDIES IN MATHEMATICS
  • 1998. A culture of proving in school mathematics? in INFORMATION AND COMMUNICATIONS TECHNOLOGIES IN SCHOOL MATHEMATICS
  • 1995-04. Moving towards a feminist epistemology of mathematics in EDUCATIONAL STUDIES IN MATHEMATICS
  • Journal

    TITLE

    Educational Studies in Mathematics

    ISSUE

    1-3

    VOLUME

    38

    Author Affiliations

    Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1023/a:1003622804002

    DOI

    http://dx.doi.org/10.1023/a:1003622804002

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1041984642


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/1302", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Curriculum and Pedagogy", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/13", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Education", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "The Open University", 
              "id": "https://www.grid.ac/institutes/grid.10837.3d", 
              "name": [
                "Centre for Mathematics Education, Open University, Walton Hall, MK7 6AA, Milton Keynes, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Mason", 
            "givenName": "John", 
            "id": "sg:person.016307220570.03", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016307220570.03"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "The Open University", 
              "id": "https://www.grid.ac/institutes/grid.10837.3d", 
              "name": [
                "Centre for Mathematics Education, Open University, Walton Hall, MK7 6AA, Milton Keynes, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Spence", 
            "givenName": "Mary", 
            "id": "sg:person.010401527770.32", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010401527770.32"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "https://doi.org/10.1080/03033910.1988.10557705", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1007372811"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.3102/0013189x018001032", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1010595714"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.3102/0013189x018001032", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1010595714"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.17763/haer.57.1.j463w79r56455411", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1013937535"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01274177", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016308008", 
              "https://doi.org/10.1007/bf01274177"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01274177", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1016308008", 
              "https://doi.org/10.1007/bf01274177"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01273662", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019229904", 
              "https://doi.org/10.1007/bf01273662"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf01273662", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1019229904", 
              "https://doi.org/10.1007/bf01273662"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-0-387-35287-9_21", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1028199870", 
              "https://doi.org/10.1007/978-0-387-35287-9_21"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep2302_5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1038145757"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep2302_5", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1038145757"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep3102_3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040284756"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep3102_3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040284756"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep3102_3", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1040284756"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15516709cog1701_7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041068368"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15516709cog1701_7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1041068368"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1016/s0364-0213(78)80052-4", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1049701106"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep3102_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1052692963"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.1207/s15326985ep3102_2", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1052692963"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.3102/00346543067001088", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1070970289"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "https://doi.org/10.3102/00346543067001088", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1070970289"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "1999-03", 
        "datePublishedReg": "1999-03-01", 
        "description": "Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam: Myself when young did eagerly frequent, Doctor and Saint and heard great argument, About it and about: but ever more Came out by the same door as in I went (p. 341). Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions \u2018come to mind\u2019. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.", 
        "genre": "research_article", 
        "id": "sg:pub.10.1023/a:1003622804002", 
        "inLanguage": [
          "en"
        ], 
        "isAccessibleForFree": false, 
        "isPartOf": [
          {
            "id": "sg:journal.1136108", 
            "issn": [
              "0013-1954", 
              "1573-0816"
            ], 
            "name": "Educational Studies in Mathematics", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1-3", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "38"
          }
        ], 
        "name": "Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment", 
        "pagination": "135-161", 
        "productId": [
          {
            "name": "readcube_id", 
            "type": "PropertyValue", 
            "value": [
              "c7fed9ac76b1f1de4186f494e2f4a93a128c6edfdd6023b980f3bcc9459aa465"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1023/a:1003622804002"
            ]
          }, 
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1041984642"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1023/a:1003622804002", 
          "https://app.dimensions.ai/details/publication/pub.1041984642"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2019-04-11T01:04", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-uberresearch-data-dimensions-target-20181106-alternative/cleanup/v134/2549eaecd7973599484d7c17b260dba0a4ecb94b/merge/v9/a6c9fde33151104705d4d7ff012ea9563521a3ce/jats-lookup/v90/0000000001_0000000264/records_8697_00000500.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "http://link.springer.com/10.1023/A:1003622804002"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1023/a:1003622804002'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1023/a:1003622804002'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1023/a:1003622804002'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1023/a:1003622804002'


     

    This table displays all metadata directly associated to this object as RDF triples.

    107 TRIPLES      21 PREDICATES      39 URIs      19 LITERALS      7 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1023/a:1003622804002 schema:about anzsrc-for:13
    2 anzsrc-for:1302
    3 schema:author N3250b02243b44ef2a162fbda48fe4fa3
    4 schema:citation sg:pub.10.1007/978-0-387-35287-9_21
    5 sg:pub.10.1007/bf01273662
    6 sg:pub.10.1007/bf01274177
    7 https://doi.org/10.1016/s0364-0213(78)80052-4
    8 https://doi.org/10.1080/03033910.1988.10557705
    9 https://doi.org/10.1207/s15326985ep2302_5
    10 https://doi.org/10.1207/s15326985ep3102_2
    11 https://doi.org/10.1207/s15326985ep3102_3
    12 https://doi.org/10.1207/s15516709cog1701_7
    13 https://doi.org/10.17763/haer.57.1.j463w79r56455411
    14 https://doi.org/10.3102/0013189x018001032
    15 https://doi.org/10.3102/00346543067001088
    16 schema:datePublished 1999-03
    17 schema:datePublishedReg 1999-03-01
    18 schema:description Knowing-to is active knowledge which is present in the moment when it is required. To try to produce knowing-to, formal education focuses on forms of knowing which are easier to teach and to test: knowing-that (factual), knowing-how (technique and skills), and knowing-why (having a story in order to structure actions and from which to reconstruct actions). Together these constitute knowing-about the subject. Expertise is demonstrated by being able to respond to assessments: to write essays and to solve routine problems. The central problem of education is that knowing-about does not in itself guarantee knowing-to, as teachers have attested throughout the ages. For example, Edward Fitzgerald (Harrison, 1937) captures it beautifully in one stanza of his purported translation of the Rubaiyat of Omar Khayyam: Myself when young did eagerly frequent, Doctor and Saint and heard great argument, About it and about: but ever more Came out by the same door as in I went (p. 341). Instead of trying to reach definitions, we illustrate distinctions amongst kinds of knowing as used by various authors in the past. Then we turn to our own experience, for it is in one's own experience that one can locate and enliven sources of metaphoric resonances and metonymic triggers which constitute understanding. Drawing on our experience we distinguish knowing-to from other forms of knowing, and explore implications of that distinction for teaching and learning mathematics. We propose that knowing-to act in the moment depends on the structure of attention in the moment, depends on what one is aware of. Educating this awareness is most effectively done by labelling experiences in which powers have been exhibited, and developing a rich network of connections and triggers so that actions ‘come to mind’. No-one can act if they are unaware of a possibility to act; no-one can act unless they have an act to perform.
    19 schema:genre research_article
    20 schema:inLanguage en
    21 schema:isAccessibleForFree false
    22 schema:isPartOf N7fa0a088735b49b7a0900a51ec82c8c7
    23 Na4e49e9bc30b49c4963679a9182effc3
    24 sg:journal.1136108
    25 schema:name Beyond mere knowledge of mathematics: The importance of knowing-to act in the moment
    26 schema:pagination 135-161
    27 schema:productId N300bded7e4f2472184cec622e7b00db3
    28 N79f9e5d600cc431caae29593b3692c83
    29 N84e46881defe4b95973ba8b5c4987799
    30 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041984642
    31 https://doi.org/10.1023/a:1003622804002
    32 schema:sdDatePublished 2019-04-11T01:04
    33 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    34 schema:sdPublisher Nf0bfa7ef1af24aa4ab75a59a8d01b4c1
    35 schema:url http://link.springer.com/10.1023/A:1003622804002
    36 sgo:license sg:explorer/license/
    37 sgo:sdDataset articles
    38 rdf:type schema:ScholarlyArticle
    39 N300bded7e4f2472184cec622e7b00db3 schema:name readcube_id
    40 schema:value c7fed9ac76b1f1de4186f494e2f4a93a128c6edfdd6023b980f3bcc9459aa465
    41 rdf:type schema:PropertyValue
    42 N3250b02243b44ef2a162fbda48fe4fa3 rdf:first sg:person.016307220570.03
    43 rdf:rest Ne1d0d6b45417439ab5a6f1dacfdb6f9f
    44 N79f9e5d600cc431caae29593b3692c83 schema:name dimensions_id
    45 schema:value pub.1041984642
    46 rdf:type schema:PropertyValue
    47 N7fa0a088735b49b7a0900a51ec82c8c7 schema:issueNumber 1-3
    48 rdf:type schema:PublicationIssue
    49 N84e46881defe4b95973ba8b5c4987799 schema:name doi
    50 schema:value 10.1023/a:1003622804002
    51 rdf:type schema:PropertyValue
    52 Na4e49e9bc30b49c4963679a9182effc3 schema:volumeNumber 38
    53 rdf:type schema:PublicationVolume
    54 Ne1d0d6b45417439ab5a6f1dacfdb6f9f rdf:first sg:person.010401527770.32
    55 rdf:rest rdf:nil
    56 Nf0bfa7ef1af24aa4ab75a59a8d01b4c1 schema:name Springer Nature - SN SciGraph project
    57 rdf:type schema:Organization
    58 anzsrc-for:13 schema:inDefinedTermSet anzsrc-for:
    59 schema:name Education
    60 rdf:type schema:DefinedTerm
    61 anzsrc-for:1302 schema:inDefinedTermSet anzsrc-for:
    62 schema:name Curriculum and Pedagogy
    63 rdf:type schema:DefinedTerm
    64 sg:journal.1136108 schema:issn 0013-1954
    65 1573-0816
    66 schema:name Educational Studies in Mathematics
    67 rdf:type schema:Periodical
    68 sg:person.010401527770.32 schema:affiliation https://www.grid.ac/institutes/grid.10837.3d
    69 schema:familyName Spence
    70 schema:givenName Mary
    71 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010401527770.32
    72 rdf:type schema:Person
    73 sg:person.016307220570.03 schema:affiliation https://www.grid.ac/institutes/grid.10837.3d
    74 schema:familyName Mason
    75 schema:givenName John
    76 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.016307220570.03
    77 rdf:type schema:Person
    78 sg:pub.10.1007/978-0-387-35287-9_21 schema:sameAs https://app.dimensions.ai/details/publication/pub.1028199870
    79 https://doi.org/10.1007/978-0-387-35287-9_21
    80 rdf:type schema:CreativeWork
    81 sg:pub.10.1007/bf01273662 schema:sameAs https://app.dimensions.ai/details/publication/pub.1019229904
    82 https://doi.org/10.1007/bf01273662
    83 rdf:type schema:CreativeWork
    84 sg:pub.10.1007/bf01274177 schema:sameAs https://app.dimensions.ai/details/publication/pub.1016308008
    85 https://doi.org/10.1007/bf01274177
    86 rdf:type schema:CreativeWork
    87 https://doi.org/10.1016/s0364-0213(78)80052-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1049701106
    88 rdf:type schema:CreativeWork
    89 https://doi.org/10.1080/03033910.1988.10557705 schema:sameAs https://app.dimensions.ai/details/publication/pub.1007372811
    90 rdf:type schema:CreativeWork
    91 https://doi.org/10.1207/s15326985ep2302_5 schema:sameAs https://app.dimensions.ai/details/publication/pub.1038145757
    92 rdf:type schema:CreativeWork
    93 https://doi.org/10.1207/s15326985ep3102_2 schema:sameAs https://app.dimensions.ai/details/publication/pub.1052692963
    94 rdf:type schema:CreativeWork
    95 https://doi.org/10.1207/s15326985ep3102_3 schema:sameAs https://app.dimensions.ai/details/publication/pub.1040284756
    96 rdf:type schema:CreativeWork
    97 https://doi.org/10.1207/s15516709cog1701_7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1041068368
    98 rdf:type schema:CreativeWork
    99 https://doi.org/10.17763/haer.57.1.j463w79r56455411 schema:sameAs https://app.dimensions.ai/details/publication/pub.1013937535
    100 rdf:type schema:CreativeWork
    101 https://doi.org/10.3102/0013189x018001032 schema:sameAs https://app.dimensions.ai/details/publication/pub.1010595714
    102 rdf:type schema:CreativeWork
    103 https://doi.org/10.3102/00346543067001088 schema:sameAs https://app.dimensions.ai/details/publication/pub.1070970289
    104 rdf:type schema:CreativeWork
    105 https://www.grid.ac/institutes/grid.10837.3d schema:alternateName The Open University
    106 schema:name Centre for Mathematics Education, Open University, Walton Hall, MK7 6AA, Milton Keynes, UK
    107 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...