Relational or operational: Primary students

understanding of the equal sign

Jodie Hunter University of

Plymouth

BSRLM November 2009

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.

Background Research

• Errors – Syntactic indicator – Operator symbol

• Computational reasoning

• Relational reasoning

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)

Data collection

• Individual interviews

- What does = mean? Can it mean anything else?

- Equivalence problems

8 + 6 = + 5 26 + = 28 + 15

13 – 7 = 11 - - 8 = 24 – 16

Results – Verbal explanation of the equal sign

• Operational explanations

• Relational explanations

Table 1: Percentage of operational or relational explanations given by students

Operational Relational

Year 3 (n=25) 80% 20%

Year 5 (n=25) 56% 44%

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

Correct Incorrect

8 + 6 = + 5 24% 76%

26 + = 28 + 15 16% 84%

13 – 7 = 11 - 40% 60%

- 8 = 24 – 16 12% 88%

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 + 5Direct sum error

88% 60%

8 + 6 = 19 + 5Sum of all error

8% 8%

Other erroneous response

4%

No response 4%

8 + 6 = 9 + 5Relational strategy

4%

8 + 6 = 9 + 5Computational strategy

24%

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 + 15Complete the sum error

96% 56%

26 + 53 = 28 + 15Direct sum error

8%

Other erroneous response

4%

No response 4%

26 + 17 = 28 + 15Relational strategy

4%

26 + 17 = 28 + 15 Computational strategy

28%

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 - 6Direct sum error

4% 16%

Other erroneous response

16% 12%

No response 80% 32%

13 - 7 = 11 - 5Relational strategy

4%

13 – 7 = 11 - 5 Computational strategy

36%

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 - 16Complete the sum error

24% 36%

8 - 18 = 24 – 16 or

6 – 18 = 24 - 16Direct sum error

8% 16%

Other error 28% 8%

No response 40% 12%

26 - 18 = 24 - 16Relational strategy

4%

10 – 18 = 24 – 16Incorrect computational strategy

16%

26 - 18 = 24 - 16Computational strategy

8%

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.