Connecting the equals sign
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Connecting the equals sign Ian Jones Æ Dave Pratt
Published online: 11 January 2007 Springer Science+Business Media B.V. 2007
Abstract Children tend to view the equals sign as an operator symbol bereft of the rich relational properties of equality statements. It has been argued by some that this restricted view of the equals sign is due to cultural or cognitive factors. We suggest a significant factor is that rich relational meanings lack relevance within the context of paper-based arithmetic. One possible way to allow learners access to relational meanings is through interaction with technologically supported utilities for the equals sign. We report upon a trial in which two students draw on existing and emerging notions of mathematical equivalence in order to connect an onscreen = object with other arithmetical objects. Keywords equals sign Æ equivalence Æ microworld Early records of the use of the equals sign appear to have been based on the notion of equivalence between two entities. The originator of the familiar horizontal parallel lines was Robert Recorde whose reasoning is revealed in The Whetstone of Witte, published in 1557: ‘‘I will sette as I doe often in woorke use, a paire of paralleles, or Gemowe [twin] lines of one length, thus: =, bicause noe .2. thynges, can be moare equalle.’’(cited in Cajori, 1923, p. 167). In modern mathematics a variety of use-meanings are in currency. Often the symbol = is used as a place-indicator for the result to a numeric calculation, as in 2 + 2 = 4 (Behr, Erlwanger, & Nichols, 1976), or to indicate that both sides have the same value, as in 2 + 2 = 3 + 1 (Kieran, 1981). Within algebra the equals sign can indicate assignment as in n = 13; or indicate sameness for all values of a variable as in 2(x + 3) = 2x + 6; or draw attention to specific values of a variable that produce equality as in x + 13 = 2x + 15 (Hewitt, 2006). Digital technology has widened the I. Jones (&) Æ D. Pratt Centre for New Technologies Research in Education (CeNTRE), The University of Warwick, Coventry CV4 7AL, UK e-mail: [email protected]
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use-meanings of the equals sign beyond those afforded by static media. On traditional calculators = appears on a button pressed to get the result to a programmed sequence of numbers and operators. Within computer programming languages = can assign values or compare two inputs and return a Boolean result. In this paper we track two students’ articulations during a series of arithmetical tasks in order to identify how they use the equals sign and how that usage evolves. We relate those changes to the novel use of technology in this experiment.
1 Children’s perceptions of the equals sign We begin by considering research claims in the literature, with emphasis on the somewhat subtle differences of focus and interpretation of data. This offers not just a backdrop for tracing children’s meanings but, as we shall see, children’s motivation for the context in which these meanings are traced. It is widely reported th
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