A Comparative Analysis of the Distributive Property in U.S. and Chinese Elementary Mathematics Textbooks

REFERENCES

• Anderson , J. R. 1993 . Rules of the mind. , Hillsdale, NJ : Lawrence Erlbaum .
   Google Scholar
• Anderson , J. R. 1995 . Cognitive psychology and its implications , 4th ed. , New York : Freeman .
   Google Scholar
• Baek , J. M. 2008 . “ Developing algebraic thinking through explorations in multiplication ” . In Algebra and algebraic thinking in school mathematics , Edited by: Greens , C. E. and Rubenstein , R. 141 – 154 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Ball , D. L. 1988 . Knowledge and reasoning in mathematical pedagogy: Examining what prospective teachers bring to teacher education (Unpublished doctoral dissertation) , East Lansing : Michigan State University .
   Google Scholar
• Ball , D. L. and Cohen , D. K. 1996 . Reform by the book: What is—Or might be—the role of curriculum materials in teaching learning and instructional reform? . Educational Researcher , 25 ( 9 ) : 6 – 8 . 14
   Google Scholar
• Bell , A. 1996 . “ Problem-solving approaches to algebra: Two aspects ” . In Approaches to algebra , Edited by: Bednarz , N. , Kieran , C. and Lee , L. 167 – 186 . Dordrecht, , The Netherlands : Kluwer Academic .
   Google Scholar
• Brown , S. W. and Bennett , E. D. 2002 . The role of practice and automaticity in temporal and nontemporal dual-task performance . Psychological Research , 66 : 80 – 89 .
   PubMed Web of Science ®Google Scholar
• Bruner , J. S. 1960/1977 . The process of education. , Cambridge, MA : Harvard University Press .
   Google Scholar
• Cai , J. 1995 . A cognitive analysis of U.S. and Chinese students’ mathematical performance on tasks involving computation, simple problem solving, and complex problem solving . Journal for Research in Mathematics Education Monograph , 7 : 1 – 160 .
   Google Scholar
• Cai , J. 2004 . Why do U.S. and Chinese students think differently in mathematical problem solving? Impact of early algebra learning and teachers’ beliefs . Journal of Mathematical Behavior , 23 : 135 – 167 .
   Google Scholar
• Cai , J. , Lew , H. C. , Morris , A. , Moyer , J. C. , Ng , S. F. and Schmittau , J. 2005 . The development of students’ algebraic thinking in earlier grades: A cross-cultural comparative perspective . ZDM: The International Journal on Mathematics Education , 37 ( 1 ) : 5 – 15 .
   Google Scholar
• Cai , J. and Moyer , M. 2008 . “ Developing algebraic thinking in earlier grades: Some insights from international comparative studies ” . In Algebra and algebraic thinking in school mathematics , Edited by: Greens , C. E. and Rubenstein , R. 169 – 180 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Carpenter , S. K. , Pashler , H. , Cepeda , N. J. and Alvarez , D. 2007 . “ Applying the principles of testing and spacing to classroom learning ” . In Proceedings of the 29th Annual Conference of the Cognitive Proceedings of the 29th Annual Conference of the Cognitive , Edited by: McNamara , D. S. and Trafton , J. G. 19 Nashville, TN : Cognitive Science Society .
   Google Scholar
• Carpenter , T. P. , Fennema , E. , Franke , M. , Levi , L. and Emspon , S. 1999 . Children's mathematics: Cognitively guided instruction , Portsmouth, NH : Heinemann .
   Google Scholar
• Carpenter , T. P. , Franke , L. P. and Levi , L. 2003 . Thinking mathematically: Integrating arithmetic & algebra in elementary school , Portsmouth, NH : Heinemann .
   Google Scholar
• Cepeda , N. J. , Pashler , H. , Vul , E. , Wixted , J. T. and Rohrer , D. 2006 . Distributed practice in verbal recall tasks: A review and quantitative synthesis . Psychological Bulletin , 132 : 354 – 380 .
   PubMed Web of Science ®Google Scholar
• Charles , R. I. , Crown , W. , Fennell , F. , Caldwell , J. H. , Cavanagh , M. Chancellor , D. 2004 . Scott Foresman–Addison Wesley mathematics , Student edition , 3 – 6 . Glenview, IL : Pearson Education . grades K–6; Teacher edition, grades
   Google Scholar
• Chi , M. T. H. 2000 . “ Self-explaining: The dual processes of generating and repairing mental models ” . In Advances in instructional psychology , Edited by: Glaser , R. 161 – 238 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Chi , M. T. H. , de Leeuw , N. , Chiu , M. and LaVancher , C. 1994 . Eliciting self-explanations improves understanding . Cognitive Science , 18 : 439 – 477 .
   Web of Science ®Google Scholar
• Chi , M. T. H. , Feltovitch , P. J. and Glaser , R. 1981 . Categorization and representation of physics problems by experts and novices . Cognitive Science , 5 : 121 – 152 .
   Web of Science ®Google Scholar
• Colhoun , J. , Gentner , D. and Loewenstein , J. 2005 . “ Learning abstract principles through principle-case comparison ” . In Proceedings of the 30th Annual Conference of the Cognitive Science Society , Edited by: Love , B. C. , McRae , K. and Sloutsky , V. M. 1659 – 1664 . Austin, TX : Cognitive Science Society .
   Google Scholar
• Craig , S. D. , Sullins , J. , Witherspoon , A. and Gholson , B. 2006 . The deep-level-reasoning-question effect: The role of dialogue and deep-level-reasoning questions during vicarious learning . Cognition and Instruction , 24 : 565 – 591 .
   Web of Science ®Google Scholar
• Gentner , D. 1983 . Structure-mapping: A theoretical framework for analogy . Cognitive Science , 7 : 155 – 170 .
   Web of Science ®Google Scholar
• Gentner , D. , Loewenstein , J. and Thompson , L. 2003 . Learning and transfer: A general role for analogical encoding . Journal of Educational Psychology , 95 : 393 – 408 .
   Web of Science ®Google Scholar
• Gentner , D. , Loewenstein , J. and Thompson , L. 2004 . “ Analogical encoding: Facilitating knowledge transfer and integration ” . In Proceedings of the 26th Annual Conference of the Cognitive Science Society , Edited by: Forbus , K. , Gentner , D. and Regier , T. 452 – 457 . Chicago, IL : Cognitive Science Society .
   Google Scholar
• Goldstone , R. L. and Sakamoto , Y. 2003 . The transfer of abstract principles governing complex adaptive systems . Cognitive Psychology , 46 : 414 – 466 .
   PubMed Web of Science ®Google Scholar
• Goldstone , R. L. and Son , J. Y. 2005 . The transfer of scientific principles using concrete and idealized simulations . Journal of the Learning Sciences , 14 : 69 – 110 .
   Web of Science ®Google Scholar
• Greenes , C. , Larson , M. , Leiva , M. A. , Shaw , J. M. , Stiff , L. , Vogeli , B. R. and Yeatts , K. 2005 . Houghton Mifflin , 3 – 6 . Boston, MA : Houghton Mifflin . Student edition grades K–6; Teacher edition, grades
   Google Scholar
• Greer , B. 1992 . “ Multiplication and division as models of situations ” . In Handbook of research on mathematics teaching and learning , Edited by: Grouws , D. A. 276 – 295 . New York : Macmillan .
   Google Scholar
• Hall , R. , Kibler , D. , Wenger , E. and Trnxaw , C. 1989 . Exploring the episodic structure of algebra story problem solving . Cognition and Instruction , 6 : 223 – 283 .
   Web of Science ®Google Scholar
• Hamann , M. S. and Ashcraft , M. H. 1986 . Textbook presentations of the basic addition facts . Cognition and Instruction , 3 : 173 – 192 .
   Google Scholar
• Hiebert , J. , Carpenter , T. P. , Fennema , E. , Fuson , K. C. , Wearne , D. Murray , H. 1997 . Making sense: Teaching and learning mathematics with understanding , Portsmouth, NH : Heinemann .
   Google Scholar
• Hiebert , J. and Wearne , D. 1996 . Instruction, understanding and skill in multidigit addition and instruction . Cognition and Instruction , 14 : 251 – 283 .
   Web of Science ®Google Scholar
• Horizon Research . 2002 . The 2000 national survey of science and mathematics education: Compendium of tables Retrieved July 15, 2008, from http://2000survey.horizon-research.com/reports/tables/tables_complete.pdf
   Google Scholar
• Howe , R. 1999 . Knowing and teaching elementary mathematics . Notices of the American Mathematical Society , 46 : 881 – 887 .
   Google Scholar
• Kaminiski , J. A. , Sloutsky , V. M. and Heckler , A. F. 2006 . “ Effects of concreteness on representation: An explanation for differential transfer ” . In Proceedings of the 28th Annual Conference Proceedings of the 28th Annual Conference , Edited by: Sun , R. and Miyake , N. 1581 – 1586 . Mahwah, NJ : Cognitive Science Society .
   Google Scholar
• Koedinger , K. R. and Terao , A. 2002 . “ A cognitive task analysis of using pictures to support pre-algebraic reasoning ” . In Proceedings of the 24th Annual Conference of the Cognitive Science Society , Edited by: Schunn , C. D. and Gray , W. 542 – 547 . Mahwah, NJ : Lawrence Erlbaum .
   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 representation on quantitative reasoning . Journal of the Learning Sciences , 13 : 129 – 164 .
   Web of Science ®Google Scholar
• Lannin , J. K. , Barker , D. D. and Townsend , B. E. 2007 . How students view the general nature of their errors . Education Studies in Mathematics , 66 : 43 – 59 .
   Google Scholar
• Li , X. , Ding , M. , Capraro , M. M. and Capraro , R. M. 2008 . Sources of differences in children's understandings of mathematical equality: Comparative analysis of teacher guides and student texts in China and in the United States . Cognition and Instruction , 26 : 195 – 217 .
   Web of Science ®Google Scholar
• Ma , L. 1999 . Knowing and teaching elementary mathematics: Understanding of fundamental mathematics in China and the United States , Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Mason , J. 2008 . “ Making use of children's powers to produce algebraic thinking ” . In Algebra in the early grades , Edited by: Kaput , J. J. , Carraher , D. W. and Blanton , M. L. 57 – 94 . New York : Lawrence Erlbaum .
   Google Scholar
• Mayer , R. E. 2001 . Multimedia learning , New York : Cambridge University Press .
   Google Scholar
• McNeil , N. M. and Alibali , M. W. 2005 . Knowledge change as a function of mathematics experience: All contexts are not created equal . Journal of Cognition and Development , 6 : 285 – 308 .
   Web of Science ®Google Scholar
• McNeil , N. M. , Grandau , L. , Knuth , E. J. , Alibali , M. W. , Stephens , A. C. Hattikudur , S. 2006 . Middle-school students’ understanding of the equal sign: The books they read can't help . Cognition and Instruction , 24 : 367 – 385 .
   Web of Science ®Google Scholar
• McNeil , N. M. and Jarvin , L. 2007 . When theories don't add up: Disentangling the manipulatives debate . Theory into Practice , 46 : 309 – 316 .
   Web of Science ®Google Scholar
• Moreno , R. and Mayer , R. C. 1999 . Multimedia-supported metaphors for meaning making in mathematics . Cognition and Instruction , 17 : 215 – 248 .
   Web of Science ®Google Scholar
• Murata , A. 2008 . Mathematics teaching and learning as a mediating process: The case of tape diagrams . Mathematical Thinking and Learning , 10 : 374 – 406 .
   Google Scholar
• Nathan , M. , Kintsch , W. and Young , E. 1992 . A theory of algebra-word-problem comprehension and its implications for the design of learning environments . Cognition and Instruction , 9 : 329 – 389 .
   Web of Science ®Google Scholar
• National Council of Teachers of Mathematics . 2000 . Principles and standards for school mathematics , Reston, VA : Author .
   Google Scholar
• National Council of Teachers of Mathematics . 2006 . Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence , Reston, VA : Author .
   Google Scholar
• National Mathematics Advisory Panel . 2008 . Foundations for success: The final report of the National Mathematics Advisory Panel , Washington, DC : U.S. Department of Education .
   Google Scholar
• National Research Council . 2001 . Adding it up: Helping children learn mathematics , Washington, DC : National Academy Press .
   Google Scholar
• National Research Council . 2004 . On evaluating curricular effectiveness: Judging the quality of K–12 mathematics evaluations , Edited by: Confrey , J. and Stohl , V. Washington, DC : National Academy Press .
   Google Scholar
• Pashler , H. , Bain , P. M. , Bottge , B. A. , Graesser , A. , Koedinger , K. McGaniel , M. 2007 . Organizing instruction and study to improve student learning , 2007 – 2004 . Washington, DC : National Center for Education Research . NCER
   Google Scholar
• Piroli , P. L. and Anderson , J. R. 1985 . The role of learning from examples in the acquisition of recursive programming . Canadian Journal of Psychology , 39 : 240 – 272 .
   Google Scholar
• Porter , A. 1989 . A curriculum out of balance: The case of elementary school mathematics . Educational Researcher , 18 ( 5 ) : 9 – 15 .
   Google Scholar
• Renkl , A. 2002 . Worked-out examples: Instructional explanations support learning by self-explanations . Learning and Instruction , 12 : 529 – 556 .
   Web of Science ®Google Scholar
• Renkl , A. , Atkinson , R. K. and Große , C. S. 2004 . How fading worked solution steps works—A cognitive load perspective . Instructional Science , 32 : 59 – 82 .
   Web of Science ®Google Scholar
• Renkl , A. , Atkinson , R. K. , Maier , U. and Staley , R. 2002 . From example study to problem solving: Smooth transitions help learning . Journal of Experimental Education , 70 : 293 – 315 .
   Web of Science ®Google Scholar
• Renkl , A. , Stark , R. , Gruber , H. and Mandl , H. 1998 . Learning from worked-out examples: The effects of example variability and elicited self-explanations . Contemporary Educational Psychology , 23 : 90 – 108 .
   PubMed Web of Science ®Google Scholar
• Resnick , L. B. , Cauzinille-Marmeche , E. and Mathieu , J. 1987 . “ Understanding algebra ” . In Cognitive processes in mathematics , Edited by: Sloboda , J. and Rogers , D. 169 – 203 . Oxford : Clarendon .
   Google Scholar
• Resnick , L. B. and Omanson , S. F. 1987 . “ Learning to understand arithmetic ” . In Advances in instructional psychology , Edited by: Glaser , R. Vol. 3 , 41 – 95 . Hillsdale, NJ : Lawrence Erlbaum .
   Google Scholar
• Rittle-Johnson , B. and Koedinger , K. 2001 . “ Using cognitive models to guide instructional design: The case of fraction division ” . In Proceedings of the 23rd Annual Conference of the Cognitive Science Society , Edited by: Moore , J. and Stenning , K. 857 – 862 . Mahwah, NJ : Lawrence Erlbaum .
   Google Scholar
• Rittle-Johnson , B. , Siegler , S. and Alibali , M. W. 2001 . Developing conceptual understanding and procedural skill in mathematics: An iterative process . Journal of Educational Psychology , 93 : 346 – 362 .
   Web of Science ®Google Scholar
• Rittle-Johnson , B. and Star , J. 2007 . Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations . Journal of Educational Psychology , 99 : 561 – 574 .
   Web of Science ®Google Scholar
• Schifter , D. 1999 . “ Reasoning about operations: Early algebraic thinking in grades K–6 ” . In Developing mathematical reasoning in grades K–12 , Edited by: Stiff , L. V. and Curcio , F. R. 62 – 81 . Reston, VA : National Council of Teachers of Mathematics .
   Google Scholar
• Schifter , D. , Monk , S. , Russell , S. J. and Bastable , V. 2008 . “ Early algebra: What does understanding the laws of arithmetic mean in the early grades? ” . In Algebra in the early grades , Edited by: Kaput , J. J. , Carraher , D. W. and Blanton , M. L. 413 – 448 . New York : Lawrence Erlbaum .
   Google Scholar
• Sheng , C. 2008 . Curriculum analysis (Grades 2–6) Retrieved October 25, 2008 from http://www.xxsx.cn/item.asp?SmallClassID=2&ItemID=571
   Google Scholar
• Schmidt , W. H. , Wang , H. C. and McKnight , C. C. 2005 . Curriculum coherence: An examination of US mathematics and science content standards from an international perspective . Journal of Curriculum Studies , 37 : 525 – 559 .
   Web of Science ®Google Scholar
• Stevenson , H. W. and Stigler , J. W. 1992 . The learning gap: Why our schools are failing and what we can learn form Japanese and Chinese education , New York : Simon & Schuster .
   Google Scholar
• Stigler , J. , Fuson , K. , Ham , M. and Kim , M . 1986 . An analysis of addition and subtraction word problems in American and Soviet elementary mathematics textbooks . Cognition and Instruction , 3 : 153 – 171 .
   Google Scholar
• Su , L. and Wang , N. 2005 . Elementary mathematics textbook , 1 – 12 . Nanjing : Jiang Su Education Press .
   Google Scholar
• Sweller , J. 1988 . Cognitive load during problem solving: Effects on learning . Cognitive Science , 12 : 257 – 285 .
   Web of Science ®Google Scholar
• Sweller , J. 1989 . Cognitive technology: Some procedures for facilitating learning and problem solving in mathematics and science . Journal of Cognitive Psychology , 81 : 457 – 466 .
   Google Scholar
• Sweller , J. 1999 . Instructional design in technical areas , Victoria, Australia : Australian Council of Education .
   Google Scholar
• Trends in International Mathematics and Science Study . 1999 . Mathematics and science achievement of eighth-graders in 1999 Retrieved June 1, 2009 from http://nces.ed.gov/timss/results99_1.asp
   Google Scholar
• Trends in International Mathematics and Science Study . 2003 . Highlights from the trends in international mathematics and science study (TIMSS) 2003 Retrieved June 1, 2009 from http://nces.ed.gov/pubs2005/2005005.pdf
   Google Scholar
• Trends in International Mathematics and Science Study . 2007 . Highlights from TIMSS 2007: Mathematics and science achievement of U.S. fourth and eighth-grade students in an international context Retrieved June 1, 2009 from http://nces.ed.gov/pubs2009/2009001.pdf.
   Google Scholar
• Uttal , D. H. , Liu , L. L. and Deloache , J. S. 1999 . “ Taking a hard look at concreteness: Do concrete objects help young children learn symbolic relations? ” . In Child psychology: A handbook of contemporary issues , Edited by: Tamis-LeMonda , C. S. 177 – 192 . Philadelphia : Psychology Press .
   Google Scholar
• VanLehn , K. , Jones , R. M. and Chi , M. T. H. 1992 . A model of the self-explanation effect . The Journal of the Learning Sciences , 2 ( 1 ) : 1 – 59 .
   Google Scholar
• Zhu , X. and Simon , H. A. 1987 . Learning mathematics from examples and by doing . Cognition and Instruction , 4 : 137 – 166 .
   Google Scholar