### Diagnostic Algebra Assessment Definitions

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Equality Symbol Misconception

As algebra teachers, we all know how frustrating it can be to teach a particular concept and to have a percentage of our students not get it. We try different approaches and activities but to no avail. These students just do not seem to grasp the concept. Often, we blame the students for not trying hard enough. Worse yet, others blame us for not teaching students well enough.

Students often learn the equality symbol misconception when they begin learning mathematics. Rather than understanding that the equal sign indicates equivalence between the expressions on the left side and the right side of an equation, students interpret the equal sign as meaning “do something” or the sign before the answer. This problem is exacerbated by many adults solving problems in the following way:
5 × 4 + 3 = ?
5 × 4 = 20 + 3 = 23
Students may also have difficulty understanding statements like 7 = 3 + 4 or 5 = 5, since these do not involve a problem on the left and an answer on the right.

Falkner presented the following problem to 6th grade classes:

8 + 4 = [] + 5

All 145 students gave the answer of 12 or 17. It can be assumed that students got 12 since 8 + 4 = 12. The 17 may be from those who continued the problem: 12 + 5 = 17.

Students with this misconception may also have difficulty with the idea that adding or subtracting the same amount from both sides of an equation maintains equality. Kieran gives this example:

Solve for x: 2x + 3 = 5 + x
2x + 3 – 3 = 5 + x
2x = 5 + xx – 3
2xx = 5 – 3
x = 2

The answer is correct, but several steps of the solution contain incorrect equations.

In summary, a student with this misconception will:

• Assume the solution to a problem is the number after the equal sign.
• Treat the equal sign as a command to do something, thus fill in a blank or variable to the right of the equal sign with the completed operation from the left hand side of the equation.
• Not add, subtract, multiply, or divide equally from both sides of an equal sign.

Graphing Misconception Definition

In algebra, students learn that a graph is a representation for a function. Students learn to translate between graphs, equations, and table of values.

But just as the translation between equations and word problems is more difficult, students sometimes find interpreting the graph of a real world situation more difficult. Students may forget the algebraic relationships they have learned and resort to graphical misconceptions. The most common graphing misconceptions are treating a graph as a picture and slope-height confusion.

An example of interpreting a graph as a picture might be a problem asking a student to draw a speed vs. time graph for a biker riding over a hill. Students with the misconception would draw the hill, and ignore that speed is asked for. Students do not look at the graph as showing speed as a function of time, but think of it more literally.

An example of slope/height confusion might use the following graph:

The question could ask "Which internet company costs more per hour at 2 hours?" A student with the misconception would choose Call.com since it costs more at 2 hours. The student does not recognize that the problem asks for slope.