Table 2.

Construct Map for Math Equivalence Knowledge

Level	Description	Equation Structure	Sample Items
4 Comparative Relational	Successful with a variety of equation structures by comparing expressions on the two sides of the equal sign. Generate a relational definition of the equal sign.	Operations on both sides with multi-digit numbers or with multiple instances of a variable.	ST	67 + 86 = 68 + 85 (true or false? explain why)
			EQ	What does the equal sign mean?
			OE	__ + 55 = 37 + 54
3 Basic Relational	Successful with equation structures with operations on both sides of the equal sign. Recognize a relational definition of the equal sign as correct.	Operations on both sides: a + b = c + d a + b – c = d + e	ST	31 + 16 = 16 + 31 (true or false?)
			EQ	Is “the same as” a good definition of equal sign?
			OE	5 + __ = 6 + 2
2 Flexible Operational	Successful with equation structures that are compatible with an operational view of the equal sign.	Operations on the right side: c = a + b No operations: a = a	ST	4 = 4 + 0 (true or false?)
			EQ	10 cents __ one dime (select correct symbol)
			OE	7 = __ + 3
1 Rigid Operational	Successful with an operations-equals-answer equation structure. Generate an operational definition of the equal sign.	Operations on the left side: a + b = c	ST	5 + 2 = 7 (true or false?)
			EQ	Identify a pair equal to 3 + 6
			OE	__ + 5 = 9

Note. Table adapted from Rittle-Johnson et al. (2011, p. 87). Success includes the ability to solve, evaluate, and encode equations of a particular structure. ST = Equation-Structure Items, EQ = Equal-Sign Items, OE = Open Equation-Solving Items.