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Equality Symbol Misconception

Graphing Misconception Definition

Concept of a Variable 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*: 2*x* + 3 = 5 + *x*

2*x* + 3 – 3 = 5 + *x*

2*x* =
5 + *x* – *x* – 3

2*x* – *x* =
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.

Concept of a Variable Misconception

Transitioning from arithmetic to algebra can be a challenging learning experience for students. One of the key reasons is the use of letters to represent variables. Even after taking an algebra course, many students do not understand the use of letters in equations and therefore can not grasp the concept of a variable.

Booth (1984) explains that letters can be interpreted
as a specific known number, as multiple values instead of one, as an object,
or simply ignored. An example of a typical mistake that students make when
they do not understand the concept of a variable is when asked to "add
4 onto 3*n*," an
answer of 3*n*4 or 7*n* is given. As another example of a
concept of a variable misconception, when students are asked to write an
equation for the following statement using the variables *S* and *P*:
"There are six times as many students as professors at this university,"
(Clement, 1982), they interpret the variable *P* as meaning professor,
instead of number of professors.