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RELATING ONE AND TWO-DIMENSIONAL QUANTITIES: STUDENTS’ MULTIPLICATIVE REASONING IN COMBINATORIAL AND SPATIAL CONTEXTS (ULN 6 7LOOHPD ,QGLDQD 8QLYHUVLW\ 3XUGXH 8QLYHUVLW\ ,QGLDQDSROLV Abstract ,Q WKLV SDSHU , RXWOLQH WKH GLVWLQFWLRQ WKDW 9HUJQDXG PDGH EHWZHHQ LVRPRUSKLVP RI PHDVXUHV SUREOHPV DQG SURGXFW RI PHDVXUHV SUREOHPV , XVH WKLV GLVWLQFWLRQ DV D UDWLRQDOH IRU FRQVLGHULQJ KRZ WKH PXOWLSOLFDWLYH VWUXFWXUHV WKDW WKUHH HLJKWK JUDGH VWXGHQWV XVHG WR VROYH &DUWHVLDQ SURGXFW SUREOHPV FRXOG EH D EDVLV IRU XQGHUVWDQGLQJ WKHLU VROXWLRQ RI SUREOHPV WKDW LQYROYHG UHODWLQJ OHQJWKV ZLGWKV DQG DUHDV %\ GRLQJ VR , GUDZ D FRQQHFWLRQ EHWZHHQ VWXGHQWV¶ FRPELQDWRULDO UHDVRQLQJ DQG KRZ WKH\ TXDQWL¿HG VLWXDWLRQV WKDW LQYROYHG WKHLU VSDWLDO DQG PXO- WLSOLFDWLYH UHDVRQLQJ 9HUJQDXG SURYLGHG D PDWKHPDWLFDO DQDO\VLV RI WKH PXOWLSOLFDWLYH FRQFHSWXDO ¿HOG LQ ZKLFK KH LGHQWL¿HG KRZ WKH PXOWLSOLFDWLYH UHDVRQLQJ LQ PDQ\ FRPELQDWRULFV SUREOHPV ZDV VLPLODU WR WKH PXOWLSOLFDWLYH UHDVRQLQJ LQYROYHG LQ HVWDEOLVKLQJ TXDQWLWDWLYH UHODWLRQVKLSV DPRQJ OHQJWKV DUHDV DQG YROXPHV LH UHODWLRQVKLSV DPRQJ JHRPHWULF TXDQWLWLHV $OWKRXJK KH SRV- LWHG WKDW WKHUH ZDV D FRQQHFWLRQ EHWZHHQ WKHVH WZR GRPDLQV RI PXOWLSOLFDWLYH UHDVRQLQJ WKH FRQ- QHFWLRQ WKDW KH SURSRVHG ZDV EDVHG RQ D PDWKHPDWLFDO DQDO\VLV RI SUREOHP VLWXDWLRQV QRW EDVHG RQ PDNLQJ PRGHOV RI VWXGHQWV¶ UHDVRQLQJ ,Q WKLV SDSHU , RXWOLQH WKH PDWKHPDWLFDO GLVWLQFWLRQV WKDW OHG 9HUJQDXG WR GUDZ KLV FRQFOXVLRQV , WKHQ SURYLGH D VXPPDU\ RI GDWD IURP D \HDU FRQ- VWUXFWLYLVW WHDFKLQJ H[SHULPHQW ,Q WKLV VXPPDU\ , ¿UVW RXWOLQH IRXU GLIIHUHQW OHYHOV RI D VFKHPH WKDW WKUHH HLJKWK JUDGH VWXGHQWV FRQVWUXFWHG ZKLOH VROYLQJ &DUWHVLDQ SURGXFW SUREOHPV $IWHU , KDYH RXWOLQHG WKHVH GLIIHUHQW OHYHOV RI WKH VFKHPH , SURYLGH DQ H[DPSOH RI KRZ VWXGHQWV ZKR ZHUH RSHUDWLQJ DW WKH WKLUG DQG IRXUWK OHYHO RI WKH VFKHPH TXDQWL¿HG VLWXDWLRQV WKDW LQYROYHG OHQJWKV DQG DUHDV GLIIHUHQWO\ Vergnaud $V SDUW RI 9HUJQDXG¶V DQDO\VLV RI WKH PXOWLSOLFDWLYH FRQFHSWXDO ¿HOG KH GLIIHUHQWLDWHG EHWZHHQ LVRPRUSKLVP RI PHDVXUHV SUREOHPV DQG SURGXFW RI PHDVXUHV SUREOHPV +H GH¿QHG LVRPRUSKLVP RI PHDVXUHV SUREOHPV DV DQ\ SUREOHP WKDW IURP DQ DGXOW¶V PDWKHPDWLFDO SHUVSHF- WLYH FDQ LQYROYH D GLUHFW SURSRUWLRQ EHWZHHQ WZR PHDVXUH VSDFHV 7KHVH SUREOHPV LQFOXGH HTXDO JURXSV SUREOHPV SDUWLWLYH DQG TXRWLWLYH GLYLVLRQ SUREOHPV UDWLR SUREOHPV GLUHFW SURSRUWLRQ SUREOHPV DQG FRQVWDQW UDWH SUREOHPV VHH DOVR 9DQ 'RRUHQ 'H %RFN -DQVVHQV 9HUVFKDIIHO 2QH H[DPSOH RI DQ LVRPRUSKLVP RI PHDVXUHV SUREOHP LV DQ HTXDO JURXSV SUREOHP OLNH WKH 'RQXW 3UREOHP The donut problem, 7KHUH DUH GRQXWV LQ D SDFNDJH -LP EX\V SDFNDJHV RI GRQXWV +RZ PDQ\ GRQXWV GLG -LP EX\" ,W LV DQ LVRPRUSKLVP RI PHDVXUHV SUREOHP EHFDXVH LW FDQ EH VHHQ DV LQYROYLQJ WZR PHDVXUH VSDFHV WKH QXPEHU RI GRQXWV DQG WKH QXPEHU RI SDFNDJHV DQG LWV VROXWLRQ FDQ LQYROYH HVWDE- OLVKLQJ D GLUHFW SURSRUWLRQ EHWZHHQ WKHVH WZR PHDVXUH VSDFHV LH GRQXWV SHU SDFNDJH GRQXWV SHU SDFNDJHV

relating one and two-dimensional quantities: students' multiplicative

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Page 1: relating one and two-dimensional quantities: students' multiplicative

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RELATING ONE AND TWO-DIMENSIONAL QUANTITIES: STUDENTS’ MULTIPLICATIVE REASONING IN

COMBINATORIAL AND SPATIAL CONTEXTS

(ULN�6��7LOOHPD,QGLDQD�8QLYHUVLW\�3XUGXH�8QLYHUVLW\�,QGLDQDSROLV

Abstract,Q� WKLV� SDSHU�� ,� RXWOLQH� WKH� GLVWLQFWLRQ� WKDW�9HUJQDXG� �������PDGH� EHWZHHQ� LVRPRUSKLVP� RI�PHDVXUHV�SUREOHPV�DQG�SURGXFW�RI�PHDVXUHV�SUREOHPV��,�XVH�WKLV�GLVWLQFWLRQ�DV�D�UDWLRQDOH�IRU�FRQVLGHULQJ� KRZ� WKH�PXOWLSOLFDWLYH� VWUXFWXUHV� WKDW� WKUHH� HLJKWK� JUDGH� VWXGHQWV� XVHG� WR� VROYH�&DUWHVLDQ�SURGXFW�SUREOHPV�FRXOG�EH�D�EDVLV�IRU�XQGHUVWDQGLQJ�WKHLU�VROXWLRQ�RI�SUREOHPV�WKDW�LQYROYHG�UHODWLQJ�OHQJWKV��ZLGWKV��DQG�DUHDV��%\�GRLQJ�VR��,�GUDZ�D�FRQQHFWLRQ�EHWZHHQ�VWXGHQWV¶�FRPELQDWRULDO�UHDVRQLQJ�DQG�KRZ�WKH\�TXDQWL¿HG�VLWXDWLRQV�WKDW�LQYROYHG�WKHLU�VSDWLDO�DQG�PXO-WLSOLFDWLYH�UHDVRQLQJ���

9HUJQDXG� ������� SURYLGHG� D�PDWKHPDWLFDO� DQDO\VLV� RI� WKH�PXOWLSOLFDWLYH� FRQFHSWXDO� ¿HOG� LQ�ZKLFK� KH� LGHQWL¿HG� KRZ� WKH�PXOWLSOLFDWLYH� UHDVRQLQJ� LQ�PDQ\� FRPELQDWRULFV� SUREOHPV�ZDV�VLPLODU�WR�WKH�PXOWLSOLFDWLYH�UHDVRQLQJ�LQYROYHG�LQ�HVWDEOLVKLQJ�TXDQWLWDWLYH�UHODWLRQVKLSV�DPRQJ�OHQJWKV��DUHDV��DQG�YROXPHV��L�H���UHODWLRQVKLSV�DPRQJ�JHRPHWULF�TXDQWLWLHV���$OWKRXJK�KH�SRV-LWHG�WKDW�WKHUH�ZDV�D�FRQQHFWLRQ�EHWZHHQ�WKHVH�WZR�GRPDLQV�RI�PXOWLSOLFDWLYH�UHDVRQLQJ��WKH�FRQ-QHFWLRQ�WKDW�KH�SURSRVHG�ZDV�EDVHG�RQ�D�PDWKHPDWLFDO�DQDO\VLV�RI�SUREOHP�VLWXDWLRQV�QRW�EDVHG�RQ�PDNLQJ�PRGHOV�RI�VWXGHQWV¶�UHDVRQLQJ��,Q�WKLV�SDSHU��,�RXWOLQH�WKH�PDWKHPDWLFDO�GLVWLQFWLRQV�WKDW�OHG�9HUJQDXG�WR�GUDZ�KLV�FRQFOXVLRQV��,�WKHQ�SURYLGH�D�VXPPDU\�RI�GDWD�IURP�D���\HDU�FRQ-VWUXFWLYLVW�WHDFKLQJ�H[SHULPHQW��,Q�WKLV�VXPPDU\��,�¿UVW�RXWOLQH�IRXU�GLIIHUHQW�OHYHOV�RI�D�VFKHPH�WKDW�WKUHH�HLJKWK�JUDGH�VWXGHQWV�FRQVWUXFWHG�ZKLOH�VROYLQJ�&DUWHVLDQ�SURGXFW�SUREOHPV��$IWHU�,�KDYH�RXWOLQHG�WKHVH�GLIIHUHQW�OHYHOV�RI�WKH�VFKHPH��,�SURYLGH�DQ�H[DPSOH�RI�KRZ�VWXGHQWV�ZKR�ZHUH�RSHUDWLQJ�DW� WKH�WKLUG�DQG�IRXUWK�OHYHO�RI�WKH�VFKHPH�TXDQWL¿HG�VLWXDWLRQV�WKDW� LQYROYHG�OHQJWKV�DQG�DUHDV�GLIIHUHQWO\���

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,W� LV�DQ� LVRPRUSKLVP�RI�PHDVXUHV�SUREOHP�EHFDXVH� LW�FDQ�EH�VHHQ�DV� LQYROYLQJ� WZR�PHDVXUH�VSDFHV��WKH�QXPEHU�RI�GRQXWV�DQG�WKH�QXPEHU�RI�SDFNDJHV��DQG�LWV�VROXWLRQ�FDQ�LQYROYH�HVWDE-OLVKLQJ�D�GLUHFW�SURSRUWLRQ�EHWZHHQ�WKHVH�WZR�PHDVXUH�VSDFHV��L�H�����GRQXWV�SHU���SDFNDJH� ����GRQXWV�SHU���SDFNDJHV���

Page 2: relating one and two-dimensional quantities: students' multiplicative

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REASONING IN COMBINATORIAL AND SPATIAL CONTEXTS

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The two-suit card problem��<RX�KDYH�WKH�$FH�WKUX�.LQJ�RI�+HDUWV²���FDUGV��<RXU�IULHQG�KDV�WKH�$FH�WKUX�.LQJ�RI�&OXEV²���FDUGV��<RX�PDNH�WZR�FDUG�KDQGV�E\�SDLU-LQJ�RQH�RI�\RXU�FDUGV�ZLWK�RQH�RI�\RXU�IULHQG¶V�FDUGV��+RZ�PDQ\�GLIIHUHQW�WZR�FDUG�KDQGV�FDQ�\RX�PDNH"

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The ice cream problem. 7KHUH�DUH���LFH�FUHDP�ÀDYRUV��+RZ�PDQ\�WZR�VFRRS�LFH�FUHDP�FRQHV�FDQ�\RX�PDNH�LI�WKH�RUGHU�RI�WKH�VFRRSV�PDWWHUV"

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Page 3: relating one and two-dimensional quantities: students' multiplicative

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