Differences in grade 7 students’ understanding of the equal sign

Figures & data

Table 1. List of assessment items.

Topic	Position	Item
Explanation of concept	Item 1	Define the equal sign (given example 3 + 4 = 7)
		Name the symbol Define the symbol (prompt for additional meaning)
Evaluation of examples	Item 6	Rate 12 given definition as the equal sign as either correct, somewhat correct, or incorrect. 3 definitions expressing an operational view 3 definitions expressing a sameness-relational view 3 definitions expressing a substitutive-relational view 3 distracter items
Application and justifications of procedures tasks	Item 2	Judge if the value of n is the same in “2 n + 15 = 31” and “2 n-9 + 15 = 31–9”
	Item 2 alt	The literal symbol n is replaced by an empty box.
	Item 3	Judge if “162 + 78 = 159 + 81” is true or false. Explain reasoning.
	Item 4	Judge the value of b in relation to a in “a = b + 5”

Table 2. Parts of the Coding Scheme.

Item		Sample response	Coding
1	Explanation of concept	“The final answer”	Operational
		“The symbol shows the answer. The answer to the problem.”	Operational
		“Instead of writing ‘answer’ you can write the symbol = ”	Operational
		“It means that both sides are equal”	Relational
		“The symbol means that what is on both side of it equals in amount”	Relational
		“Equal value on both sides”	Relational
2–3	Application and justifications on procedure tasks	“It’s at the end of the second: 31–9 which is 22”	Answer after equal sign
	Explanations	“Added 162 + 78 and the answer was not 159”	Answer after equal sign
		“I use the problem backwards 31–15 = 16 and the I figure out the number in the box by 2x … =16”	Solve and compare
		“I added. Both give the same answer.”	Solve and compare
		“The problems look the same if you remove −9 from both sides. If you remove the same amount from both sides of = the value doesn’t change.”	Recognize equivalence
		“I compare. 162 and 159 have 3 between them and 78 and 81 have 3 between them.”	Recognize equivalence

Table 3. Students’ acceptance of understanding in items 6.

Understanding	Reject, N = 159	Accept, N =159	Mean	Std.Dev.
Operational	18%	67%	4.30	1.444
Sameness- relational	20%	72%	4.47	1.799
Substitutive- relational	43%	46%	2.97	2.219

Note. Because student overall scores of 3 (=neutral) are not included, percentages for each understanding do not add up to 100%.

Table 4. Results of the Principal Component Analysis (PCA).

The symbol “ = ” means	Predicted understanding	Loading on Factor 1	Loading on Factor 2	Loading on Factor 3	Loading on Factor 4
… the answer to the problem	Operational	−0.180	0.756	0.052	−0.109
… the total	Operational	−0.141	0.745	−0.078	−0.016
… work out the results	Operational	−0.243	0.415	−0.498	−0.081
… the something is equal to another thing	Sameness- relational	0.036	−0.051	0.284	0.764
… that two amounts are the same	Sameness- relational	0.356	−0.147	−0.022	0.830
… that both sides have the same value	Sameness- relational	0.537	−0.009	0.085	0.756
… that one side can replace the other	Substitutive- relational	0.834	−0.181	−0.032	0.125
… that the right side can be swapped for the left	Substitutive- relational	0.785	−0.084	0.059	0.359
… that two sides can be exchanged	Substitutive- relational	0.857	−0.169	0.093	0.285
… the start of the problem	Distracter	−0.134	0.101	−0.790	−0.178
… to repeat the numbers	Distracter	0.113	0.034	−0.711	−0.095
… the end of the problem	Distracter	−0.029	0.616	−0.300	−0.040

Note. Rotation method: Oblimin with Kaiser Normalization. Loadings >.400 are shown in bold.

Table 5. Nonparametric correlations between results from items 1 and 6 (Spearman’s rho).

	Provide a relational best definition (item 1)	Accept operational understanding (item 6)	Accept sameness/ relational understanding (item 6)
Provide a relational best definition
Accept operational understanding (item 6)	−.326**
Accept sameness- relational understanding (item 6)	.529**	−.158*
Accept substitutive- relational understanding (item 6)	.421**	−.247	.427**

**Correlation is significant at the 0.01 level (2-tailed).

**Correlation is significant at the 0.05 level (2-tailed).

Table 6. Nonparametric correlations between results from items 2–4, 1 and 6 (Spearman’s rho).

	Total score, items 2–4
Provide a relational best definition (item 1)	.312**
Accept operational understanding (item 6)	−.141
Accept sameness-relational understanding (item 6)	.336**
Accept substitutive- relational understanding (item 6)	.264**

**Correlation is significant at the 0.01 level (2-tailed)

Table 7. Explanations for correct and incorrect answers to items 2–3 (collapsed, N = 138).

		Don’t know/ No answer	Answer after equal sign	Solve and compare	Recognize equivalence
Incorrect	Count % within incorrect	33 31%	17 16%	57 53%	1 1%
Correct	Count % within correct	6 3%	0 0%	165 79%	39 19%