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Relational or operational: Primary students understanding of the equal sign Jodie Hunter University of Plymouth BSRLM November 2009

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Background Research Well-documented difficulties in developing student understanding of algebraic concepts. Need for students to understand the equal sign as a representation of an equivalence relationship. This paper will examine student understanding of the equal sign through their verbal explanations and their attempts to solve equivalence problems.

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Background Research Errors –Syntactic indicator –Operator symbol Computational reasoning Relational reasoning

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Study Context Initial data collection for a year long design experiment Urban primary school 25 Year 3 students (7-8 years old) 25 Year 5 students (9-10 years old)

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Data collection Individual interviews - What does = mean? Can it mean anything else? - Equivalence problems 8 + 6 = + 526 + = 28 + 15 13 – 7 = 11 - - 8 = 24 – 16

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Results – Verbal explanation of the equal sign Operational explanations Relational explanations Table 1: Percentage of operational or relational explanations given by students OperationalRelational Year 3 (n=25)80%20% Year 5 (n=25)56%44%

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Results – Equivalence problems All Year 3 students gave incorrect answers for the four equivalence problems Table 2: Percentage of Year 5 students (n=25) who gave correct / incorrect responses for the equivalence problems CorrectIncorrect 8 + 6 = + 5 24%76% 26 + = 28 + 15 16%84% 13 – 7 = 11 - 40%60% - 8 = 24 – 16 12%88%

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Results – Responses to equivalence problems Table 2: Percentage of student responses to 8 + 6 = + 5 Year 3 (n=25)Year 5 (n=25) 8 + 6 = 14 + 5 Direct sum error 88%60% 8 + 6 = 19 + 5 Sum of all error 8% Other erroneous response 4% No response4% 8 + 6 = 9 + 5 Relational strategy 4% 8 + 6 = 9 + 5 Computational strategy 24%

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Results – Responses to equivalence problems Table 2: Percentage of student responses to 26 + = 28 + 15 Year 3 (n=25)Year 5 (n=25) 26 + 2 = 28 + 15 Complete the sum error 96%56% 26 + 53 = 28 + 15 Direct sum error 8% Other erroneous response 4% No response4% 26 + 17 = 28 + 15 Relational strategy 4% 26 + 17 = 28 + 15 Computational strategy 28%

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Results – Responses to equivalence problems Table 2: Percentage of student responses to 13 – 7 = 11 - Year 3 (n=25)Year 5 (n=25) 13 - 7 = 11 - 6 Direct sum error 4%16% Other erroneous response 16%12% No response80%32% 13 - 7 = 11 - 5 Relational strategy 4% 13 – 7 = 11 - 5 Computational strategy 36%

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Results – Responses to equivalence problems Table 2: Percentage of student responses to – 18 = 24 - 16 Year 3 (n=25)Year 5 (n=25) 42 - 18 = 24 - 16 Complete the sum error 24%36% 8 - 18 = 24 – 16 or 6 – 18 = 24 - 16 Direct sum error 8%16% Other error28%8% No response40%12% 26 - 18 = 24 - 16 Relational strategy 4% 10 – 18 = 24 – 16 Incorrect computational strategy 16% 26 - 18 = 24 - 16 Computational strategy 8%

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Conclusion and implications Lack of understanding of the equal sign as equivalence. Some improvement between Year 3 and Year 5 students. Need for specific attention to the equal sign and use of relational strategies.

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