Examples of Misconceptions

Below is a short list of the many misconceptions students have in mathematics. To begin exploring student cognition, it is essential that the teacher understand how the student reasons and what the student does and does not understand.

Multiplication always makes smaller. Division always makes larger.

When solving word problems with numbers less than 1, students often use the wrong operation.

  • See the following articles:

    Tirosh, Dina: Graeber, Anna O.. "Evoking Cognitive Conflict to Explore Preservice Teachers' Thinking about Division." Journal for Research in Mathematics Education 21 (Mar 1990) p 98-108.

    Graeber, Anna O.. "Misconceptions about Multiplication and Division." Arithmetic Teacher 40 (Mar 1993) p 408-411.

    The authors suggest the misconception comes from several sources:

    The authors suggest using the area interpretation of multiplication and the measurement interpretation of division. How many .25s are in 2? How many 1/4 inches are in 2 inches?

    Scaling Errors

    Students often think that when you double all dimensions of an object that you multiply its crossectional area and its volume by a factor of 2, not 4 and 8 respectively.

  • Show them the formulas. A=lw and V=lwh. When you multiply l and w each by 2 you multiply A by 4. Have them calculate an example of the area of a rectangle.

  • Give them a sheet of graph paper. Have them outline a rectangle. Then have them outline a rectangle with double the dimensions. Count blocks. Then subdivide the large rectangle into four of the smaller ones.

  • Order them a 10" pizza and a 14" pizza. Cut it in a way to show that the large has twice the pizza as the small. Bon Appetit!

    Freshman Exponentiation

    (a+b)p is not equal to ap + bp
    In particular students like to "simplify" (x2+1)½ to (x+1) immediately.

  • The Pythagorean Theorem says:
    Given a right triangle with legs of length a and b and hypotenuse of length c that a2+b2=c2.
    If a2+b2=(a+b)2 then a+b would equal c.

  • Graphing exercises:
    y=x2+1 and y=(x+1)2 are different parabolas.
    x2+y2=1 and (x+y)2=1 have totally different shapes.

  • The Binomial Theorem

    Constant of Integration

    Many students think that when integrating f to get F+c that c=F(0).

  • A particle travels around a circular track.
    Its x-velocity as a function of time is given by x'(t)=sin(t).
    Its initial position is given by x(0)=1.
    Integrating x' yields x(t)=-cos(t)+c. But c=2 not 1.

    Division by Zero

    When solving equations like x3+2x2+x=0 many students immediately divide out an x and then forget about it.

  • This third degree polynomial has three roots. If you divide by x the resulting polynomial has only two roots.

  • Remind students that they can NOT divide by zero. In this case, the function x can take on the value zero. If they factor they will not lose a solution because they are only changing the form of the equation not the set of solutions.

    A wonderful source for many more examples:

    Davis, R. B. (1984). Learning mathematics The cognitive science approach to mathematics education. Norwood, NJ Ablex.