Math Monsters - Math Misconceptions

 Misconceptions in mathematics usually arise from previous inadequate teaching, informal thinking, or poor remembrance. Each one of us lives with a unique set of beliefs and ways of problem solving. That is true in case of life and is equally true in case of Mathematics. Students do not come to the classroom as "blank slates" (Resnick, 1983). Rather, they come with informal theories constructed from everyday experiences. These theories have been actively constructed. They provide an everyday functionality to make sense of the world but are often incomplete half-truths (Mestre, 1987). They are misconceptions.

Now the basic reason to focus our attention on these misconceptions is the fact that once rooted in the student’s memory, misconceptions are hard to erase. Misconceptions are firmly held, descriptive, and explanatory systems for scientific and logico-mathematical phenomena, that is, systems of belief about mathematics. And the problem with these beliefs is that these differ from what is incorporated into the standard curriculum. Research suggests that certain constellations of these belief systems show remarkable consistency across ages, abilities, and nationalities.

Moreover most of the times, belief systems are resistant to change through traditional instruction. Repeating a lesson or making it clearer will not help students who base their reasoning on strongly held misconceptions. (Champagne, Klopfer & Gunstone, 1982; McDermott, 1984; Resnick, 1983). Students tend to be emotionally and intellectually attached to their misconceptions, partly because they have actively constructed

them and partly because they give ready methodologies for solving various problems. They

definitely interfere with learning when students use them to interpret new experiences.

Thus it becomes extremely important to recognize student misconceptions and to re-educate students tocorrect mathematical thinking.

How to re-educate away from misconceptions?

Changing the conceptual framework of students is one of the keys goals in repairing

mathematics and science misconceptions. That is to say, it is not usually successful to merely

inform (e.g. lecture) the student on a misconception. The misconception must be changed

internally partly through the student’s belief systems and partly through their own cognition.

Lochead & Mestre (1988) describe an effective inductive technique for these purposes. There are three steps.

1. Probe for and determine qualitative understanding.

2. Probe for and determine quantitative understanding.

3. Probe for and determine conceptual reasoning.

In addition, it is helpful to confront students with counterexamples to their misconceptions. A

self-discovered counterexample will have a far stronger and lasting effect. Incorrect beliefs

can be loosened somewhat when so confronted.

Misconceptions are but one facet of faulty, inaccurate, or incorrect thinking. These are all intertwined causing students unlimited trouble in grasping with mathematics from the most elementary concepts through calculus. Student misconceptions cause teachers immense frustration about why their teaching isn’t "getting through."

In my next post, I attempt to consider student misconceptions in mathematics, particularly those that impact algebra and algebraic thinking.