The Effects of Presenting Multidigit Mathematics Problems in a Realistic Context on Sixth Graders' Problem Solving

REFERENCES

• Abedi , J. and Hejri , F. 2004 . Accommodations for students with limited English proficiency in the National Assessment of Educational Progress . Applied Measurement in Education , 17 : 371 – 392 .
   Web of Science ®Google Scholar
• Abedi , J. and Lord , C. 2001 . The language factor in mathematics tests . Applied Measurement in Education , 14 : 219 – 234 .
   Web of Science ®Google Scholar
• Adams , R. J. , Wilson , M. and Wang , W. C. 1997 . The multidimensional random coefficients multinomial logit model . Applied Psychological Measurement , 21 : 1 – 23 .
   Web of Science ®Google Scholar
• Adams , R. J. and Wu , M. L. 2000 . PISA 2000 technical report , Paris , , France : OECD .
   Google Scholar
• Albert , J. H. 1992 . A Bayesian estimation of normal ogive item response curves using Gibbs sampling . Journal of Educational Statistics , 17 : 251 – 269 .
   Google Scholar
• Ambrose , R. , Baek , J.-M. and Carpenter , T. P. 2003 . “ Children's invention of multidigit multiplication and division algorithms ” . In The development of arithmetic concepts and skills: Constructing adaptive expertise , Edited by: Baroody , A. J. and Dowker , A. 305 – 336 . Mahwah , NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Anghileri , J. 1989 . An investigation of young children's understanding of multiplication . Educational Studies in Mathematics , 20 : 367 – 385 .
   Google Scholar
• Baroody , A. J. and Dowker , A. 2003 . The development of arithmetic concepts and skills: Constructing adaptive expertise , NJ : Lawrence Erlbaum Associates . Mahwah
   Google Scholar
• Bates , D. and Maechler , M. 2010 . lme4: Linear mixed modeling using S4 classes [Computer program and manual]. Retrieved from http://cran.r-project.org/web/packages/lme4/index.html
   Google Scholar
• Béguin , A. A. and Glas , C. A. W. 2001 . MCMC estimation and some model-fit analysis of multidimensional IRT models . Psychometrika , 66 : 541 – 562 .
   Web of Science ®Google Scholar
• Beishuizen , M. 1993 . Mental strategies and materials or models for addition and subtraction up to 100 in Dutch second grades . Journal for Research in Mathematics Education , 24 : 294 – 323 .
   Web of Science ®Google Scholar
• Beishuizen , M. , Van Putten , C. M. and Van Mulken , F. 1997 . Mental arithmetic and strategy use with indirect number problems up to one hundred . Learning and Instruction , 7 : 87 – 106 .
   Web of Science ®Google Scholar
• Berends , I. E. and Van Lieshout , E. C. D. M. 2009 . The effect of illustrations in arithmetic problem-solving: Effects of increased cognitive load . Learning and Instruction , 19 : 345 – 353 .
   Web of Science ®Google Scholar
• Blöte , A. , Van der Burg , E. and Klein , A. S. 2001 . Students' flexibility in solving two-digit addition and subtraction problems: Instruction effects . Journal of Educational Psychology , 93 : 627 – 638 .
   Web of Science ®Google Scholar
• Buijs , C. 2008 . Leren vermenigvuldigen met meercijferige getallen , Utrecht , , The Netherlands : Freudenthal Institute for Science and Mathematics Education . [Learning to multiply with multidigit numbers]
   Google Scholar
• Carpenter , T. P. , Moser , J. M. and Romberg , T. A. , eds. 1982 . Addition and subtraction: A cognitive perspective , Hillsdale , NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Carr , M. and Davis , H. 2001 . Gender differences in arithmetic strategy use: A function of skill and preference . Contemporary Educational Psychology , 26 : 330 – 347 .
   PubMed Web of Science ®Google Scholar
• Carr , M. and Jessup , D. L. 1997 . Gender differences in first-grade mathematics strategy use: Social and metacognitive influences . Journal of Educational Psychology , 89 : 318 – 328 .
   Web of Science ®Google Scholar
• Carraher , T. N. , Carraher , D. W. and Schliemann , A. D. 1985 . Mathematics in the streets and in schools . British Journal of Developmental Psychology , 3 : 21 – 29 .
   Web of Science ®Google Scholar
• CITO . 2009 . Terugblik en resultaten 2009. Eindtoets basisonderwijs , Arnhem , , The Netherlands : Author . [Retrospect and results of the End of Primary School Test 2009]
   Google Scholar
• Cohen , J. 1960 . A coefficient of agreement for nominal scales . Educational and Psychological Measurement , 20 : 37 – 46 .
   Web of Science ®Google Scholar
• Cummins , D. D. , Kintsch , W. , Reusser , K. and Weimer , R. 1988 . The role of understanding in solving word problems . Cognitive Psychology , 20 : 405 – 438 .
   Web of Science ®Google Scholar
• De Boeck , P . 2008 . Random item models . Psychometrika , 73 : 533 – 559 .
   Web of Science ®Google Scholar
• De Boeck , P. , Bakker , M. , Zwitser , R. , Nivard , M. , Hofman , A. , Tuerlinckx , F. and Partchev , I. 2011 . The estimation of item response models with the lmer function from the lme4 package in R . Journal for Statistical Software , 39 ( 12 ) : 1 – 28 .
   PubMed Web of Science ®Google Scholar
• De Boeck , P. and Wilson , M. 2004 . Explanatory item response models: A generalized linear and nonlinear approach , New York , NY : Springer .
   Google Scholar
• De Corte , E. , Verschaffel , L. and De Win , L. 1985 . Influence of rewording verbal problems on children's problem representations and solutions . Journal of Educational Psychology , 77 : 460 – 470 .
   Web of Science ®Google Scholar
• Dowker , A. 2005 . Individual differences in arithmetic. Implications for psychology, neuroscience and education , Hove , , UK : Psychology Press .
   Google Scholar
• Embretson , S. E. and Reise , S. P. 2000 . Item response theory for psychologists , Mahwah , NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Fischer , G. H. 1987 . Linear logistic test model as an instrument in educational research . Acta Psychologica , 37 : 359 – 374 .
   Google Scholar
• Freudenthal , H. 1973 . Mathematics as an educational task , Dordrecht , , The Netherlands : Reidel .
   Google Scholar
• Freudenthal , H. 1991 . Revisiting mathematics education. China lectures , Dordrecht , , The Netherlands : Kluwer Academic Publishers .
   Google Scholar
• Fuchs , L. S. , Fuchs , D. , Compton , D. L. , Powell , S. R. , Seethaler , P. M. , Capizzi , A. M. and Fletcher , J.M. 2006 . The cognitive correlates of third-grade skill in arithmetic, algorithmic computation, and arithmetic word problems . Journal of Educational Psychology , 98 : 29 – 43 .
   Web of Science ®Google Scholar
• Fuchs , L. S. , Fuchs , D. , Stuebing , K. , Fletcher , J. M. , Hamlett , C. L. and Lambert , W. 2008 . Problem solving and computational skill: Are they shared or distinct aspects of mathematical cognition? . Journal of Educational Psychology , 100 : 30 – 47 .
   PubMed Web of Science ®Google Scholar
• Gallagher , A. M. , De Lisi , R. , Holst , P. C. , McGillicuddy-De Lisi , A. V. , Morely , M. and Cahalan , C. 2000 . Gender differences in advanced mathematical problem solving . Journal of Experimental Child Psychology , 75 : 165 – 190 .
   PubMed Web of Science ®Google Scholar
• Gravemeijer , K. 1997 . Commentary. Solving word problems: A case of modelling? . Learning and Instruction , 7 : 389 – 397 .
   Web of Science ®Google Scholar
• Gravemeijer , K. and Doorman , M. 1999 . Context problems in realistic mathematics education: A calculus course as an example . Educational Studies in Mathematics , 39 : 111 – 128 .
   Google Scholar
• Greer , B. 1997 . Modelling reality in mathematics classrooms: The case of word problems . Learning and Instruction , 7 : 293 – 307 .
   Web of Science ®Google Scholar
• Hartzel , J. , Agresti , A. and Caffo , B. 2001 . Multinomial logit random effects models . Statistical Modelling , 1 : 81 – 102 .
   Web of Science ®Google Scholar
• Hickendorff , M. 2011 . “ Explanatory latent variable modeling of mathematical ability in primary school: Crossing the border between psychometrics and psychology (Unpublished doctoral dissertation) ” . Leiden , , The Netherlands : Leiden University .
   Google Scholar
• Hickendorff , M. in press . The language factor in elementary mathematics: computational skills and applied problem solving in a multidimensional IRT framework . Applied Measurement in Education. ,
   Google Scholar
• Hickendorff , M. , Heiser , W. J. , Van Putten , C. M. and Verhelst , N. D. 2009 . Solution strategies and achievement in Dutch complex arithmetic: Latent variable modeling of change . Psychometrika , 74 : 331 – 350 .
   Google Scholar
• Hickendorff , M. , Van Putten , C. M. , Verhelst , N. D. and Heiser , W. J. 2010 . Individual differences in strategy use on division problems: Mental versus written computation . Journal of Educational Psychology , 102 : 439 – 452 .
   Google Scholar
• Hop , M. 2012 . Balans van het reken-wiskundeonderwijs halverwege de basisschool 5 , Arnhem , , The Netherlands : CITO . [Fifth assessment of mathematics education halfway primary school]
   Google Scholar
• Imbo , I. and Vandierendonck , A. 2007 . Do multiplication and division strategies rely on executive and phonological working memory resources? . Memory & Cognition , 35 : 1759 – 1771 .
   PubMed Web of Science ®Google Scholar
• Janssen , R. , Schepers , J. and Peres , D. 2004 . “ Models with item and item group predictors ” . In Explanatory item response models: A generalized linear and nonlinear approach , Edited by: Boeck , P. D. and Wilson , M. 189 – 212 . New York , NY : Springer .
   Google Scholar
• Janssen , J. , Van der Schoot , F. and Hemker , B. 2005 . Balans van het reken-wiskundeonderwijs aan het einde van de basisschool 4 , Arnhem , , The Netherlands : CITO . [Fourth assessment of mathematics education at the end of primary school]
   Google Scholar
• Kilpatrick , J. , Swafford , J. and Findell , B. 2001 . Adding it up. Helping children learn mathematics , Washington , DC : National Academy Press .
   Google Scholar
• Kintsch , W. and Greeno , J. G. 1985 . Understanding and solving word arithmetic problems . Psychological Review , 92 : 109 – 129 .
   PubMed Web of Science ®Google Scholar
• Koedinger , K. R. , Alibali , M. W. and Nathan , M. J. 2008 . Trade-offs between grounded and abstract representations: Evidence from algebra problem solving . Cognitive Science , 32 : 366 – 397 .
   PubMed Web of Science ®Google Scholar
• Koedinger , K. R. and Nathan , M. J. 2004 . The real story behind story problems: Effects of representations on quantitative reasoning . The Journal of the Learning Sciences , 13 : 129 – 164 .
   Web of Science ®Google Scholar
• Kuss , O. and McLerran , D. 2007 . A note on the estimation of the multinomial logistic model with correlated responses . Computer Methods and Programs in Biomedicine , 87 : 262 – 269 .
   PubMed Web of Science ®Google Scholar
• Lemaire , P. and Siegler , R. S. 1995 . Four aspects of strategic change: Contributions to children's learning of multiplication . Journal of Experimental Psychology: General , 124 : 83 – 97 .
   PubMed Web of Science ®Google Scholar
• Mabbott , D. J. and Bisanz , J. 2003 . Developmental change and individual differences in children's multiplication . Child Development , 74 : 1091 – 1107 .
   PubMed Web of Science ®Google Scholar
• Mulligan , J. T. and Mitchelmore , M. C. 1997 . Young children's intuitive models of multiplication and division . Journal for Research in Mathematics Education , 28 : 309 – 330 .
   Web of Science ®Google Scholar
• Mullis , I. V. S. , Martin , M. O. and Foy , P. 2008 . TIMSS 2007 international mathematics report. Findings from IEA's Trends in International Mathematics and Science Study at the fourth and eighth grades , Boston , MA : Boston College, TIMSS & PIRLS International Study Center .
   Google Scholar
• National Council of Teachers of Mathematics . 1989 . Curriculum and evaluation standards for school mathematics , Reston , VA : NCTM .
   Google Scholar
• National Council of Teachers of Mathematics . 2000 . Principles and standards for school mathematics , Reston , VA : NCTM .
   Google Scholar
• Organization for Economic Co-operation and Development (OECD) . 2010 . PISA 2009 results: What students know and can do. Student performance in reading, mathematics and science (Vol. 1) , Paris , , France : OECD .
   Google Scholar
• Prenger , J. 2005 . Taal telt! Een onderzoek naar de rol van taalvaardigheid en tekstbegrip in het realistische rekenonderwijs , The Netherlands : University of Groningen . [Language counts! A study into the role of linguistic skill and text comprehension in realistic mathematics education] (Doctoral dissertation)
   Google Scholar
• R Development Core Team . 2009 . R: A language and environment for statistical computing Vienna , , Austria [Computer software manual] Author. Retrieved from http://www.R-project.org
   Google Scholar
• Reckase , M. D. 2009 . Multidimensional item response theory , New York , NY : Springer .
   Google Scholar
• Riley , M. S. , Greeno , J. G. and Heller , J. J. 1983 . “ Development of children's problem solving ability in arithmetic ” . In The development of mathematical thinking , Edited by: Ginsburg , H. P. 153 – 196 . New York , NY : Academic Press .
   Google Scholar
• Rizopoulos , D. 2006 . ltm: An R package for latent variable modeling and Item Response Theory analyses . Journal of Statistical Software , 17 ( 5 ) : 1 – 25 .
   Web of Science ®Google Scholar
• Robinson , K. M. , Arbuthnott , K. D. , Rose , D. , McCarron , M. C. , Globa , C. A. and Phonexay , S. D. 2006 . Stability and change in children's division strategies . Journal of Experimental Child Psychology , 93 : 224 – 238 .
   PubMed Web of Science ®Google Scholar
• Royal Netherlands Academy of Arts and Sciences . 2009 . Rekenonderwijs op de basisschool. Analyse en sleutels tot verbetering , Amsterdam , , The Netherlands : KNAW . [Mathematics education in primary school. Analysis and recommendations for improvement]
   Google Scholar
• Scheltens , F. , Hemker , B. and Vermeulen , J. 2013 . Balans van het reken-wiskundeonderwijs aan het einde van de basisschool 5 , Arnhem , , The Netherlands : CITO . [Fifth assessment of mathematics education at the end of primary school]
   Google Scholar
• Schoenfeld , A. H. 1991 . “ On mathematics as sense-making: An informal attack on the unfortunate divorce of formal and informal mathematics ” . In Informal reasoning and education , Edited by: Voss , J. F. , Perkins , D. N. and Segal , J. W. 311 – 343 . Hillsdale , NJ : Erlbaum .
   Google Scholar
• Selter , C. 2001 . Addition and subtraction of three-digit numbers: German elementary children's success, methods and strategies . Educational Studies in Mathematics , 47 : 145 – 173 .
   Google Scholar
• Sherin , B. and Fuson , K. 2005 . Multiplication strategies and the appropriation of computational resources . Journal for Research in Mathematics Education , 36 : 347 – 395 .
   Web of Science ®Google Scholar
• Siegler , R. S. 1988a . Individual differences in strategy choices: good students, not-so-good students, and perfectionists . Child Development , 59 : 833 – 851 .
   PubMed Web of Science ®Google Scholar
• Siegler , R. S. 1988b . Strategy choice procedures and the development of multiplication skill . Journal of Experimental Psychology: General , 117 : 258 – 275 .
   PubMed Web of Science ®Google Scholar
• Siegler , R. S. and Lemaire , P. 1997 . Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method . Journal of Experimental Psychology: General , 126 : 71 – 92 .
   PubMed Web of Science ®Google Scholar
• Timmermans , R. E. , Van Lieshout , E. C. D. M. and Verhoeven , L. 2007 . Gender related effects of contemporary math instruction for low performers on problem solving behavior . Learning and Instruction , 17 : 42 – 54 .
   Web of Science ®Google Scholar
• Torbeyns , J. , De Smedt , B. , Ghesquière , P. and Verschaffel , L. 2009 . Jump or compensate? Strategy flexibility in the number domain up to 100 . ZDM Mathematics Education , 41 : 581 – 590 .
   Google Scholar
• Torbeyns , J. , Ghesquière , P. and Verschaffel , L. 2009 . Efficiency and flexibility of indirect addition in the domain of multi-digit subtraction . Learning and Instruction , 19 : 1 – 12 .
   Web of Science ®Google Scholar
• Torbeyns , J. , Verschaffel , L. and Ghesquière , P. 2004 . Strategy development in children with mathematical disabilities: Insights from the choice/no-choice method and the chronological-age/ability-level-match design . Journal of Learning Disabilities , 37 : 119 – 131 .
   PubMed Web of Science ®Google Scholar
• Torbeyns , J. , Verschaffel , L. and Ghesquière , P. 2005 . Simple addition strategies in a first-grade class with multiple strategy instruction . Cognition and Instruction , 23 : 1 – 21 .
   Web of Science ®Google Scholar
• Torbeyns , J. , Verschaffel , L. and Ghesquière , P. 2006 . The development of children's adaptive expertise in the number domain 20 to 100 . Cognition and Instruction , 24 : 439 – 465 .
   Web of Science ®Google Scholar
• Treffers , A. 1987 . Integrated column arithmetic according to progressive schematisation . Educational Studies in Mathematics , 18 : 125 – 145 .
   Google Scholar
• Treffers , A. 1993 . Wiscobas and Freudenthal: Realistic mathematics education . Educational Studies in Mathematics , 25 : 89 – 108 .
   Google Scholar
• Van den Boer , C . 2003 . Als je begrijpt wat ik bedoel. Een zoektocht naar verklaringen van achterblijvende prestaties van allochtone leerlingen in het wiskundeonderwijs , Utrecht , , The Netherlands : CD-β press . [If you get what I mean. A search for explanations of lagging achievement of non-native students in mathematics education]
   Google Scholar
• Van den Heuvel-Panhuizen , M. , Buys , K. and Treffers , A. 2001 . Kinderen leren rekenen. Tussendoelen annex leerlijnen: Hele getallen bovenbouw basisschool , Groningen , , The Netherlands : Wolters-Noordhoff . [Children learn mathematics]
   Google Scholar
• Van den Heuvel-Panhuizen , M. , Robitzsch , A. , Treffers , A. and Köller , O. 2009 . Large-scale assessments of change in student achievement: Dutch primary school students' results on written division in 1997 and 2004 as an example . Psychometrika , 74 : 351 – 365 .
   Web of Science ®Google Scholar
• Van Putten , C. M. , Van den Brom-Snijders , P. A. and Beishuizen , M. 2005 . Progressive mathematization of long division strategies in Dutch primary schools . Journal for Research in Mathematics Education , 36 : 44 – 73 .
   Web of Science ®Google Scholar
• Verhelst , N. D. and Glas , C. A. W. 1995 . “ The one-parameter logistic model ” . In Rasch models. Foundations, recent developments and applications , Edited by: Fischer , G. H. and Molenaar , I. 215 – 237 . New York , NY : Springer .
   Google Scholar
• Vermeer , H. J. , Boekaerts , M. and Seegers , G. 2000 . Motivational and gender differences: Sixth-grade students' mathematical problem solving behavior . Journal of Educational Psychology , 92 : 308 – 315 .
   Web of Science ®Google Scholar
• Verschaffel , L. , Greer , B. and De Corte , E. 2000 . Making sense of word problems , Lisse , , The Netherlands : Swets and Zeitlinger .
   Google Scholar
• Verschaffel , L. , Greer , B. and De Corte , E. 2007 . “ Whole number concepts and operations ” . In Second handbook of research on mathematics teaching and learning , Edited by: Lester , F. K. 557 – 628 . Greenwich , CT : Information Age Publishing .
   Google Scholar
• Vicente , S. , Orrantia , J. and Verschaffel , L. 2007 . Influence of situational and conceptual rewording on word problem solving . British Journal of Educational Psychology , 77 : 829 – 848 .
   PubMed Web of Science ®Google Scholar
• Wu , M and Adams , R. 2006 . Modelling mathematics problem solving item responses using a multidimensional IRT model . Mathematics Education Research Journal , 18 : 93 – 113 .
   Google Scholar