Teaching the substitutive conception of the equals sign

References

• ACME . 2007 . Position statement on functional mathematics . London : Advisory Committee on Mathematics Education .
   Google Scholar
• Askew , M. , Hodgen , J. , Hossain , S. and Bretscher , N. 2010 . Values and variables: Mathematics education in high-performing countries , London : Nuffield Foundation .
   Google Scholar
• Baroody , A.J. and Ginsburg , H.P. 1983 . The effects of instruction on children's understanding of the “equals” sign . The Elementary School Journal , 84 ( 2 ) : 199 – 212 . doi: 10.1086/461356
   Web of Science ®Google Scholar
• Behr , M. , Erlwanger , S. and Nichols , E. 1976 . How children view equality sentences. Report no. PMDC-TR-3 , Florida : Florida State University .
   Google Scholar
• Capraro , R.M. , Capraro , M.M. , Yetkiner , Z.E. , Özel , S. , Kim , H.G. and KüçüK , A.R. 2010 . An international comparison of Grade 6 students’ understanding of the equal sign . Psychological Reports , 106 ( 1 ) : 49 – 53 . doi: 10.2466/pr0.106.1.49-53
   PubMed Web of Science ®Google Scholar
• Carpenter , T.P. , Franke , M.L. and Levi , L. 2003 . Thinking mathematically: Integrating arithmetic and algebra in the elementary school , Portsmouth , NH : Heinemann .
   Google Scholar
• Children, Schools and Families Committee . 2010 . From Baker to Balls: The foundations of the education system. House of Commons ninth report of session 2009–10 . London : The Stationary Office Ltd .
   Google Scholar
• Cobb , P. 1987 . An investigation of young children's academic arithmetic contexts . Educational Studies in Mathematics , 18 ( 2 ) : 109 – 24 . doi: 10.1007/BF00314722
   Google Scholar
• Denmark , T. , Barco , B. and Voran , J. 1976 . Final report: A teaching experiment on equality , Florida : Florida State University .
   Google Scholar
• Jones , I. 2008 . A diagrammatic view of the equals sign: Arithmetical equivalence as a means, not an end . Research in Mathematics Education , 10 ( 2 ) : 151 – 65 . doi: 10.1080/14794800802233688
   Google Scholar
• Jones , I. , Inglis , M. and Gilmore , C. 2011 . Imperative and punctuative operational conceptions of the equals sign . Proceedings of the British Society for Research into Learning Mathematics , 31 ( 1 ) : 79 – 84 .
   Google Scholar
• Jones , I. , Inglis , M. , Gilmore , C. and Dowens , M. 2012 . Substitution and sameness: Two components of a relational conception of the equals sign . Journal of Experimental Child Psychology , 113 ( 1 ) : 166 – 76 . doi: 10.1016/j.jecp.2012.05.003
   PubMed Web of Science ®Google Scholar
• Jones , I. and Pratt , D. 2012 . A substituting meaning for the equals sign in arithmetic notating tasks . Journal for Research in Mathematics Education , 43 ( 1 ) : 2 – 33 . doi: 10.5951/jresematheduc.43.1.0002
   Web of Science ®Google Scholar
• Kieran , C. 1981 . Concepts associated with the equality symbol . Educational Studies in Mathematics , 12 ( 3 ) : 317 – 26 . doi: 10.1007/BF00311062
   Google Scholar
• Knuth , E.J. , Stephens , A.C. , McNeil , N. and Alibali , M.W. 2006 . Does understanding the equal sign matter? Evidence from solving equations . Journal for Research in Mathematics Education , 36 ( 4 ) : 297 – 312 .
   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 the United States . Cognition and Instruction , 26 ( 2 ) : 195 – 217 . doi: 10.1080/07370000801980845
   Web of Science ®Google Scholar
• McNeil , N. 2004 . You'll see what you mean: Students encode equations based on their knowledge of arithmetic . Cognitive Science , 283 ( 3 ) : 451 – 66 .
   Google Scholar
• McNeil , N. 2007 . U-shaped development in math: 7-year-olds outperform 9-year-olds on equivalence problems . Developmental Psychology , 43 ( 3 ) : 687 – 95 . doi: 10.1037/0012-1649.43.3.687
   PubMed Web of Science ®Google Scholar
• McNeil , N. 2008 . Limitations to teaching children 2+2=4: Typical arithmetic problems can hinder learning of mathematical equivalence . Child Development , 79 ( 5 ) : 1524 – 37 . doi: 10.1111/j.1467-8624.2008.01203.x
   PubMed Web of Science ®Google Scholar
• McNeil , N. and Alibali , M.W. 2002 . “ A strong schema can interfere with learning: The case of children's typical addition schema ” . In Proceedings of the Twenty-Fourth Annual Conference of the Cognitive Science Society , Edited by: Schunn , C. and Gray , W. 661 – 6 . Mahwah , NJ : Lawrence Erlbaum Associates .
   Google Scholar
• McNeil , N. and Alibali , M.W. 2005 . Why won't you change your mind? Knowledge of operational patterns hinders learning and performance on equations . Child Development , 76 ( 4 ) : 883 – 99 . doi: 10.1111/j.1467-8624.2005.00884.x
   PubMed Web of Science ®Google Scholar
• McNeil , N. , Fyfe , E.R. , Petersen , L.A. , Dunwiddie , A.E. and Brletic-Shipley , H. 2011 . Benefits of practicing 4=2+2: Nontraditional problem formats facilitate children's understanding of mathematical equivalence . Child Development , 82 ( 5 ) : 1620 – 33 . doi: 10.1111/j.1467-8624.2011.01622.x
   PubMed Web of Science ®Google Scholar
• McNeil , N. , Grandau , L. , Knuth , E.J. , Alibali , M.W. , Stephens , A.C. , Hattikudur , S. and Krill , D.E. 2006 . Middle-school students’ understanding of the equal sign: The books they read can't help . Cognition and Instruction , 24 ( 3 ) : 367 – 85 . doi: 10.1207/s1532690xci2403_3
   Web of Science ®Google Scholar
• Molina , M. , Castro , Encarnación and Castro , Enrique . 2009 . Elementary students’ understanding of the equal sign in number sentences . Electronic Journal of Research in Educational Psychology , 7 ( 1 ) : 341 – 68 .
   Google Scholar
• Mullis , I.V.S. , Martin , M.O. , Foy , P. , Olson , J.F. , Preuschoff , C. , Erberber , E. , Arora , A. and Galia , J. 2008 . TIMSS 2007 International mathematics report: Findings from IEA's trends in international mathematics and science study at the fourth and eighth grades , Chestnut Hill , MA : TIMSS and PIRLS International Study Center, Boston College .
   Google Scholar
• NCETM . 2009 . Mathematics matters: Final report . London : National Centre for Excellence in the Teaching of Mathematics .
   Google Scholar
• Nunes , T. , Bryant , P. and Watson , A. 2009 . Key understandings in mathematics learning , London : The Nuffield Foundation .
   Google Scholar
• Ofsted . 2008 . Mathematics: Understanding the score . London : Office for Standards in Education .
   Google Scholar
• Pirie , S. and Martin , L. 1997 . The equation, the whole equation and nothing but the equation! One approach to the teaching of linear equations . Educational Studies in Mathematics , 34 ( 2 ) : 159 – 81 . doi: 10.1023/A:1003051829991
   Google Scholar
• QCA . 2004 . Key Stage 2 Mathematics Test B Levels 3–5 . http://www.emaths.co.uk/KS2SAT.htm
   Google Scholar
• Renwick , E.M. 1932 . Children's misconceptions concerning the symbols for mathematical equality . British Journal of Educational Psychology , 2 ( 2 ) : 173 – 83 . doi: 10.1111/j.2044-8279.1932.tb02743.x
   Google Scholar
• Rittle-Johnson , B. and Alibali , M.W. 1999 . Conceptual and procedural knowledge of mathematics: Does one lead to the other? . Journal of Educational Psychology , 91 ( 1 ) : 175 – 89 . doi: 10.1037/0022-0663.91.1.175
   Web of Science ®Google Scholar
• Rittle-Johnson , B. , Matthews , P. , Taylor , R.S. and McEldoon , K.L. 2011 . Assessing knowledge of mathematical equivalence: A construct-modeling approach . Journal of Educational Psychology , 103 ( 1 ) : 85 – 104 . doi: 10.1037/a0021334
   Web of Science ®Google Scholar
• Sáenz-Ludlow , A. and Walgamuth , C. 1998 . Third graders’ interpretations of equality and the equal symbol . Educational Studies in Mathematics , 35 ( 2 ) : 153 – 87 . doi: 10.1023/A:1003086304201
   Google Scholar
• Seo , K.H. and Ginsburg , H.P. 2003 . “You've got to carefully read the math sentence …”: Classroom context and children's interpretations of the equals sign ” . In The Development of Arithmetic Concepts and Skills: Constructing Adaptive Expertise , Edited by: Baroody , A.J. and Dowker , A. 161 – 87 . Mahwah , NJ : Lawrence Erlbaum Associates .
   Google Scholar
• Sherman , J. and Bisanz , J. 2009 . Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives . Journal of Educational Psychology , 101 ( 1 ) : 88 – 100 . doi: 10.1037/a0013156
   Web of Science ®Google Scholar
• Wyse , D. and Torrance , H. 2009 . The development and consequences of national curriculum assessment for primary education in England . Educational Research , 51 ( 2 ) : 213 – 28 . doi: 10.1080/00131880902891479
   Web of Science ®Google Scholar