Students’ understanding of algebraic notation: A semiotic systems perspective

Alibali, 2007, A longitudinal examination of middle school students’ understandings of the equal sign and performance solving equivalent equations, Mathematical Thinking and Learning, 9, 221, 10.1080/10986060701360902 Arzarello, 2009, Gestures as semiotic resources in the mathematics classroom, Educational Studies in Mathematics, 70, 97, 10.1007/s10649-008-9163-z Bardini, 2004, Teaching linear functions in context with graphics calculators: students’ responses and the impact of the approach on their use of algebraic symbols, International Journal of Science and Mathematics Education, 2, 353, 10.1007/s10763-004-8075-3 Bartolini Bussi, 2008, Semiotic mediation in the mathematics classroom: artefacts and signs after a Vygotskian perspective, 720 Behr, 1980, How children view the equals sign, Mathematics Teaching, 92, 13 Berger, 2004, The functional use of a mathematical sign, Educational Studies in Mathematics, 55, 81, 10.1023/B:EDUC.0000017672.49486.f2 Blanton, 2003, Developing elementary teachers’ algebra eyes and ears, Teaching Children Mathematics, 10, 70, 10.5951/TCM.10.2.0070 Blanton, 2015, The development of children’s algebraic thinking: the impact of a comprehensive early algebra intervention in third grade, Journal for Research in Mathematics Education, 46, 39, 10.5951/jresematheduc.46.1.0039 Boaler, J. (2003). Studying and capturing the complexity of practice—the case of the dance of agency. In N. Pateman, B. Dougherty, & J. Zilliox (Eds.), Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, Honolulu, Hawaii, Vol. IV, 3–16. Brizuela, 2008, Multiple notational systems and algebraic understandings: the case of the best deal problem, 273 Brizuela, 2004, Fourth graders solving linear equations, For the Learning of Mathematics, 24, 33 Brizuela, 2004, Ten-year-old students solving linear equations, For the Learning of Mathematics, 24, 33 Brizuela, 2015, Children’s use of variables and variable notation to represent their algebraic ideas, Mathematical Thinking and Learning, 17, 34, 10.1080/10986065.2015.981939 Carraher, 2007, Early algebra and algebraic reasoning, 669 Carraher, 2006, Arithmetic and algebra in early mathematics education, Journal for Research in Mathematics Education, 37, 87 Carraher, 2008, Early algebra is not the same as algebra early, 235 Chalouh, 1988, Teaching algebraic expressions in a meaningful way Christou, 2007, Students’ interpretations of literal symbols in algebra, 283 Clement, 1981, Translation difficulties in learning mathematics, American Mathematical Monthly, 88, 286, 10.1080/00029890.1981.11995253 Clement, 1982, Algebra word problem solutions: thought processes underlying a common misconception, Journal for Research in Mathematics Education, 13, 16, 10.2307/748434 Dufour-Janvier, 1987, Pedagogical considerations concerning the problem of representation, 109 Duval, 1995, Semiosis et pensée humaine Duval, 1999, Representation, vision and visualization: Cognitive functions in mathematical thinking. Basic issues for learning, Vol. 1, 3 Duval, 2006, A cognitive analysis of problems of comprehension in a learning of mathematics, Educational Studies in Mathematics, 61, 103, 10.1007/s10649-006-0400-z Ernest, 2006, A semiotic perspective of mathematical activity: the case of number, Educational Studies in Mathematics, 61, 67, 10.1007/s10649-006-6423-7 Fairclough, 1995 Filloy, 1989, Solving equations: the transition from arithmetic to algebra, For the Learning of Mathematics, 9, 19 Fisher, 1988, The students-and-professors problem revisited, Journal for Research in Mathematics Education, 19, 260, 10.2307/749069 Godfrey, 2008, Student perspectives on equation: the transition from school to university, Mathematics Education Research Journal, 20, 71, 10.1007/BF03217478 Godino, 2010, Why is the learning of elementary arithmetic concepts difficult? Semiotic tools for understanding the nature of mathematical objects, Educational Studies in Mathematics, 77, 247, 10.1007/s10649-010-9278-x Goetz, 1984 Graham, 2000, Building a versatile understanding of algebraic variables with a graphic calculator, Educational Studies in Mathematics, 41, 265, 10.1023/A:1004094013054 Gutierrez, 1995, Unpackaging academic discourse, Discourse Processes, 19, 21, 10.1080/01638539109544903 Herscovics, 1994, A cognitive gap between arithmetic and algebra, Educational Studies in Mathematics, 27, 59, 10.1007/BF01284528 Hoffman, 2006, What is a semiotic perspective, and what could it be? Some comments on the contributions to this special issue, Educational Studies in Mathematics, 61, 279, 10.1007/s10649-006-1456-5 Küchemann, 1978, Children's understanding of numerical variables, Mathematics in School, 7, 23 Kaput, 1983, Errors in translations to algebraic equations: roots and implications, Focus on Learning Problems in Mathematics, 5, 63 Kieran, 1990, Cognitive processes involved in learning school algebra, 96 Knuth, 2005, Middle school students’ understanding of core algebraic concepts: equality & variable, Zentralblatt für Didaktik der Mathematik, 37, 68, 10.1007/BF02655899 Knuth, 2006, Does understanding the equal sign matter? Evidence from solving equations, Journal for Research in Mathematics Education, 37, 297 Lacampagne, 1995 Leont’ev, 1981, The problem of activity in psychology, 37 MacGregor, 1997, Students' understanding of algebraic notation: 11–15, Educational Studies in Mathematics, 33, 1, 10.1023/A:1002970913563 MacGregor, 2001, Does learning algebra benefit most people?, Vol. 1, 296 Maschietto, 2009, Working with artefacts: gestures, drawings and speech in the construction of the mathematical meaning of the visual pyramid, Educational Studies in Mathematics, 70, 143, 10.1007/s10649-008-9162-0 Matthews, 2012, Measure for measure: what combining diverse measures reveals about children's understanding of the equal sign as an indicator of mathematical equality, Journal for Research in Mathematics Education, 43, 316, 10.5951/jresematheduc.43.3.0316 McNeil, 2006, Middle-school students’ understanding of the equal sign: the books they read can’t help, Cognition and Instruction, 24, 367, 10.1207/s1532690xci2403_3 McNeil, 2010, A is for apple: mnemonic symbols hinder students’ interpretation of algebraic expressions, Journal of Educational Psychology, 102, 625, 10.1037/a0019105 Mehan, 1979 Mestre, 1988, The role of language comprehension in mathematics and problem solving, 201 Morgan, 2006, What does social semiotics have to offer mathematics education research?, Educational Studies in Mathematics, 61, 219, 10.1007/s10649-006-5477-x National Council of Teachers of Mathematics, 2000 National Research Council, 1998 Peirce, C. S. (1958). Collected papers of Charles Sanders Peirce. Volumes I–VI. C. Hartshorne and P. Weiss (Eds.) 1931–1935; volumes VII–VIII, A. W. Burks (Ed.) 1958. Cambridge, MA.: Harvard University Press. Philipp, 1992, The many uses of algebraic variables, The Mathematics Teacher, 85, 557, 10.5951/MT.85.7.0557 Prediger, 2010, How to develop mathematics-for-teaching and understanding: the case of meanings of the equal sign, Journal of Mathematics Teacher Education, 13, 73, 10.1007/s10857-009-9119-y RAND Mathematics Study Panel, 2003 Radford, 2007, Syntax and meaning as sensuous, visual: historical forms of algebraic thinking, Educational Studies in Mathematics, 66, 145, 10.1007/s10649-006-9024-6 Radford, 2000, Signs and meanings in students’ emergent algebraic thinking: a semiotic analysis, Educational Studies in Mathematics, 42, 237, 10.1023/A:1017530828058 Radford, 2003, Gestures, speech, and the sprouting of signs, Mathematical Thinking and Learning, 5, 37, 10.1207/S15327833MTL0501_02 Radford, 2006, The anthropology of meaning, Educational Studies in Mathematics, 61, 39, 10.1007/s10649-006-7136-7 Radford, 2008, The ethics of being and knowing: towards a cultural theory of learning, 215 Radford, 2010, Algebraic thinking from a cultural semiotic perspective, Research in Mathematics Education, 12, 1, 10.1080/14794800903569741 Rosnick, 1980, Learning without understanding: the effect of tutoring strategies on algebra misconceptions, Journal of Mathematical Behavior, 3, 3 Rotman, 2006, Toward a semiotics of mathematics, 97 Sáenz-Ludlow, 2006, Classroom interpreting games with an illustration, Educational Studies in Mathematics, 61, 183, 10.1007/s10649-006-5760-x Santi, G., & Sbaragli, S. (2008). Misconceptions and semiotics: a comparison. In A. Gagatsis (2008). Research in Mathematics Education. Conference of five cities: Nicosia, Rhodes, Bologna, Palermo, Locarno. Nicosia, Cyprus, 2008. Santi, 2011, Objectification and semiotic function, Educational Studies in Mathematics, 77, 285, 10.1007/s10649-010-9296-8 Schliemann, 2007 Schoenfeld, 1988, On the meaning of variable, Mathematics Teacher, 81, 420, 10.5951/MT.81.6.0420 Sfard, 2005, Why cannot children see as the same what grown-ups cannot see as different?: early numerical thinking revisited, Cognition and Instruction, 23, 237, 10.1207/s1532690xci2302_3 Sfard, 1998, On two metaphors for learning and on the dangers of choosing just one, Educational Researcher, 27, 4, 10.3102/0013189X027002004 Sfard, 2007, When the rules of discourse change, but nobody tells you: making sense of mathematics learning from a commognitive standpoint, The Journal of the Learning Sciences, 16, 567, 10.1080/10508400701525253 Sfard, 2012, Introduction: developing mathematical discourse—some insights from communicational research, International Journal of Educational Research, 51–52, 1, 10.1016/j.ijer.2011.12.013 Steinberg, 1991, Algebra students’ knowledge of equivalence of equations, Journal for Research in Mathematics Education, 22, 112, 10.2307/749588 Thompson, P. (1999). Representation and evolution: A discussion of Duval’s and Kaput’s papers. In F. Hitt, & M. Santos (Eds.), Proceedings of the 21 st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 1, 49–54. Trigueros, M., & Ursini, S. (1999). Does the understanding of variable evolve through schooling? In O. Zaslavsky (Ed.), Proceedings of the 23rd International Conference for the Psychology of Mathematics Education, Vol. 4, Haifa, Israel, 273–280. Trigueros, 2001, Approaching the study of algebra through the concept of variable Trigueros, 2003, Starting college students’ difficulties in working with different uses of variable, Vol. 5, 1 Usiskin, 1988, Conceptions of school algebra and uses of variables, 8 Vygotsky, 1978, Mind in society van Oers, 2000, The appropriation of mathematical symbols. A psycho-semiotic approach to mathematics learning, 133 Weinberg, A. (2009). Students’ mental models for comparison word problems. In S. L. Sward, D. W. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the 31 st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Vol. 5, 709–717. Weinberg, 2010, Undergraduate students’ interpretations of the equals sign Wertsch, 1985 White, 1996, Conceptual knowledge in introductory calculus, Journal for Research in Mathematics Education, 27, 79, 10.2307/749199