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

2. 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.

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

4. 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)

5. 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

6. 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%

7. 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%

8. 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%

9. 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%

10. 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%

11. 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%

12. 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.