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From arithmetical thought to algebraic thought: The role of the “variable”

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Abstract

The introduction of the concept of the variable represents a critical point in the arithmetic–algebraic transition. This concept is complex because it is used with different meanings in different situations. Its management depends on the particular way of using it in problem-solving. The aim of this paper was to analyse whether the notion of “unknown” interferes with the interpretation of the variable “in functional relation” and the kinds of languages used by the students in problem-solving. We also wanted to study the concept of the variable in the process of translation from algebraic language into natural language. We present two experimental studies. In the first one, we administered a questionnaire to 111 students aged 16–19 years. Drawing on the conclusions of this research we carried out the second study with two pairs of students aged 16–17 years.

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Notes

  1. According to Collins English Dictionary (1991), variable is: “an expression that can be assigned any of set of values. (As modifier) able to take any of a range of values: a variable sum. A symbol, such as x, y or z, representing an unspecified member of a class of objects, numbers etc. Variables may be used either existentially or universally: in elementary algebra variables occur in conditional equations representing unknown quantities of which the values are to be found….”. Unknown is: “a variable, or the quantity it represents, whose value is to be discovered by solving an equation; a variable in a conditional equation”. Freudenthal (1983) also groups under the name variable: the unknown, the variable in functional relation and the “polyvalent names”, that is, those that can take a set of different values, but that do not vary according to a functional dependence.

  2. Nesselman (1842) individualises three periods in the evolution of algebraic symbolism: the rhetorical phase—anterior to Diophantus in Alexandria (250 AD), in which natural language was used exclusively, without resorting to any signs; the syncopated phase—from Diophantus up to the end of the 16th century, in which some abbreviations for the unknown and the relations in more frequent use were introduced, but the calculations were performed in natural language; and the symbolic phase—introduced by Viète (1540–1603), in which letters are used for all the quantities and signs to represent the operations, symbolic language is utilised not only to solve equations but also to demonstrate general rules.

  3. The digital technologies (computer algebra systems, graphic symbolic calculators, microworlds etc.) play a very important role in students’ development of at least one kind of use of algebraic language in general and the idea of the variable in particular (Hoyles, Noss, & Adamson 2002; Lagrange and Chiappini 2007 etc.). A lot of research confirms this viewpoint, but in this paper we chose not to include the dimension of digital technologies. It will be the aim of future research.

  4. By situation/problem we mean a learning situation in which: there are initial data that specify the context and that are useful for solving the problem; there is an aim to pursue that makes sense of the mobilisation and the organisation of the things learned; there are some obligations where the obstacles to be overcome require a reorganisation of a student’s knowledge and that bring him/her to finding other means and therefore to new learning; the procedure and the solution are not evident—the student must do some research to know how to proceed.

  5. In Italian we used the term “determinare”, in the sense of “determine through the calculation” or “calculate”.

  6. The “a priori analysis” is the analysis of the “Epistemological Representations”, “Historical-epistemological Representations” and “Supposed Behaviours”, correct and incorrect, to solve a given didactic situation. The epistemological representations are the representations of the possible cognitive paths regarding a particular concept. Such representations can be prepared by a beginner subject or by a scientific community in a specific historical period. Historical-epistemological representations are the representations of the possible cognitive paths regarding the syntactic, semantic and pragmatic reconstruction of a specific concept. The supposed behaviours of students in facing the situation/problem are all the possible strategies of the solution, both correct and incorrect. Among the erroneous strategies, those that can become correct strategies will be taken into account (Spagnolo 2006).

  7. In particular, Spagnolo et al. (2008) demonstrate the theoretical and methodological questions in didactic research. They also introduce experimental works that show the effectiveness of some of the new methodologies used in this article (supplementary variable etc.).

  8. The supplementary variables represent different cognitive styles and they allow the experimental data to be analysed in a better way. The introduction of supplementary variables as ideal individuals was used in numerous experimental works of the GRIM: Spagnolo (1998, 2006, 2008). These works led us to validate this method both experimentally and theoretically. In this context, with the high number of variables in play, such an investigative method allows better highlighting of the fundamental characteristics of the a priori analysis.

  9. The chief experimental variables AL2, AL4, AL4.1 and ALb.2, which define this procedure, have greater frequencies than the variables of the other two procedures (see the table of frequencies in Appendix 2).

  10. The experimental variable AL4 “he/she adds a datum” considers two possibilities: equal winnings (AL4.1 and AL4.2) or equal bets (AL4.3). This is equivalent to introducing a new equation and to forming a system of two linear equations: “The winnings of € 300 are divided in half” (AL4.1)—equivalent to the system 3x + 4y = 300 and 3x = 4y = 150; “The winnings are equal to € 300 for both the teenagers” (AL4.2)—corresponds to the system 3x = 300 and 4y = 300; “The bets are equal” (AL4.3)—is equivalent to the system 3x + 4y = 300 and x = y.

  11. With this statistical methodology, e.g. the implication AL2→AL4.1 of 99% is interpreted in this way: if there is AL2 in the student’s procedure, then with 99% probability the procedure will also include AL4.1.

  12. The purely syntactic manipulation of the equation does not lead to the correct solution of the problem; therefore, we use the term “pseudo-algebraic”.

  13. The supplementary variables represent different cognitive styles and they allow the experimental data to be analysed in a better way. The introduction of supplementary variables as ideal individuals was used in numerous experimental works of the GRIM: Spagnolo (1998, 2006, 2008). These works led us to validate this method both experimentally and theoretically. In this context, with the high number of variables in play, such an investigative method allows better highlighting of the fundamental characteristics of the a priori analysis.

  14. In this study we prefer to use the term “multiple solutions” rather than “infinite solutions”, because we have not considered the possible connotations of the word “infinite”. However, we defined the experimental variable ALb4 to take into account cases in which the student explicitly considers the existence of infinite solutions.

  15. According to Radford (2002), sometimes the students’ signs (in this case the minus sign) constitute short scripts recounting salient parts of the original story problem (told in natural language). This author considers that, for some students, the minus sign in the expression x-2 does not perform a subtraction from the unknown x, but it is an orienting mark of a short script about the story problem. Then Vita and Alessandra had the necessity to interpret the minus sign, because this exigency is linked to the possibility of conferring to the expression the correct algebraic meaning (it is different from the scripts’ meaning).

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Acknowledgements

The authors wish to thank Prof. Luis Radford for his opportune and useful remarks on the doctoral thesis: The concept of variable in the passage from arithmetic language to algebraic language in different semiotic contexts. This paper is inspired by the experimental studies carried out in this thesis.

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Authors and Affiliations

  1. G.R.I.M. (Group of Research in Mathematics Education), Department of Mathematics, University of Palermo, Palermo, Italy

    Elsa Malisani & Filippo Spagnolo

Authors
  1. Elsa Malisani
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  2. Filippo Spagnolo
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Corresponding author

Correspondence to Filippo Spagnolo.

Additional information

This research was carried out in the different semiotic contexts of Algebra and Analytic Geometry

Appendices

Appendix 1

We determined the principal experimental variables from an a priori analysis (the complete table is in Malisani 2006, pp. 114–115). They were the following for the first query:

  1. AL1:

    The student answers the question.

  2. AL2:

    He/she shows a procedure in natural language.

  3. AL3:

    He/she shows a procedure by trial and error in natural language and/or in a half formalised language.

  4. AL4:

    He/she adds a datum.

  1. AL4.1:

    He/she adds a datum, but he/she considers that the winnings are divided in half.

  2. AL4.2:

    He/she adds a datum, but he/she considers that the winnings of the two teenagers are equal to € 300.

  3. AL4.3:

    He/she adds a datum, but he/she considers that the bets are equal.

  4. AL5:

    He/she translates the problem into a first degree equation with two unknowns.

  5. AL6:

    He/she explicitly considers the bounds of the problem.

  6. AL7:

    He/she translates the problem into a first degree equation with two unknowns and he/she applies an incorrect method where he/she writes one variable in function of the other. Then he/she replaces this variable in the original equation and thus he/she obtains an identity. In short, the student applies the method of substitution used to solve systems of equations to a single equation.

  7. AL9:

    He/she abandons the pseudo-algebraic procedure and he/she tries another method.

  8. AL11:

    He/she considers, in an explicit or implicit way, the variables of the problem in a functional relation.

  9. AL13:

    He/she makes some errors in the resolution of the equation and he/she finds (or he/she tries to find) only one solution.

  10. AL14:

    He/she considers that a relation of proportionality exists between x and y.

  11. ALb1:

    The student calculates the solution set.

  12. ALb2:

    He/she shows a particular solution of the equation.

  13. ALb3:

    He/she shows several solutions of the equation.

  14. ALb4:

    He/she considers infinite solutions expressly.

  15. ALb5:

    He/she explicitly considers that the data are insufficient to determine only one solution.

  16. ALb6:

    He/she considers multiple solutions (includes ALb4 and ALb5).

Second query::

  1. IAL1:

    The student answers the question.

  2. IAL2:

    He/she transforms the equation into its explicit form.

  3. IAL3:

    He/she solves the equation by applying an incorrect method where he/she writes one variable in function of the other.

  4. IAL4:

    He/she shows a particular solution of the equation.

  5. IAL5:

    He/she shows several solutions of the equation.

  6. IAL6:

    He/she adds another equation and forms a system.

  7. IAL7:

    He/she produces a text considering only constants.

  8. IAL7.1:

    The question refers to the second member of the equation, that is, to 18.

  9. IAL11:

    He/she answers correctly.

  10. IAL12:

    He/she produces a text considering two variables, but it does not translate the given equation exactly.

  11. IAL14:

    He/she translates algebraic language with difficulty.

Fourth query:

  1. GAa1:

    The student answers the question.

  2. GAbc1:

    The student calculates the solution set.

  3. GAbc2:

    He/she shows a particular solution of the equation.

  4. GAbc3:

    He/she shows several solutions of the equation.

  5. GAbc4:

    He/she considers multiple solutions.

Appendix 2

Table 2 shows the frequencies of the variables.

Table 2 Frequencies
Full size table

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Malisani, E., Spagnolo, F. From arithmetical thought to algebraic thought: The role of the “variable”. Educ Stud Math 71, 19–41 (2009). https://doi.org/10.1007/s10649-008-9157-x

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