What is a misconception?

Student thinking consists of many things. Formulas, relevance, tedium, and enjoyment are part of their attitudes and thinking about mathematics. One problem that leads to very serious learning difficulties in mathematics is those misconceptions student may have from previous inadequate teaching, informal thinking, or poor remembrance. It may be best to begin with a definition. From the Encarta online dictionary, a misconception is “a mistaken idea or view resulting from a misunderstanding of something." Paraphrasing from the educational literature [Pines, 1985] , we offer, 

Certain conceptual relations that are acquired may be inappropriate within a certain context. We terms such relations as "misconceptions." A misconception does not exist independently, but is contingent upon a certain existing conceptual framework. Misconceptions can change or disappear with the framework changes. 

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. In another misconceptions framework, we may say many 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. 

In this article, we consider student misconceptions in mathematics, particularly those that impact algebra and algebraic thinking. Yet, 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. In turn, student misconceptions cause teachers immense frustration about why their teaching isn’t "getting through."

Our first thought would be that misconceptions, once rooted in the student’s memory, are hard to erase. The situation is somewhat more complex. Researchers’ interest in student conceptions has been provoked by numerous studies indicating that-

1. Before formal study, persons have firmly held, descriptive, and explanatory systems for scientific and logico-mathematical phenomena, that is, systems of belief about mathematics.

2. These systems of belief differ from what is incorporated into the standard curriculum.

3. Certain constellations of these belief systems show remarkable consistency across ages, abilities, and nationalities.

4. Belief systems are resistant to change through traditional instruction. See (Champagne,Gunstone, & Klopfer, 1983; Osborne & Wittrock, 1983) See also (Confrey, Jere, 1990).

This research also suggests that 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. It is very important to recognize student misconceptions and to re-educate students to correct mathematical thinking.

Although the results apply more to children younger than high school age, Ginsburg [1977] offers a number of observations about errors:

1. Errors result from organized strategies and rules.

2. Faulty rules underlying errors have sensible origins.

3. Too often children see arithmetic as an activity isolated from their ordinary concerns. (As you will note below, many misconceptions and faulty thinking in algebra are related to misconceptions and faulty thinking with arithmetic (e.g. fractions).

4. Children often demonstrate a gap between formal and informal knowledge.

The last point on formal vs. informal knowledge requires definition. Usually, formal knowledge refers to that which is taught in an organized, structured, educational institution. It refers to a system of interrelated definitions and proofs, experiments and arguments. It usually is linked with written methods. On the other hand, informal knowledge refers to more tentative intuitive conjectures and mental strategies. Informal knowledge is generated or learned through one’s personal actions. That is, informal knowledge refers to routines that are carried out mechanically, or by habit, or by tradition.

A body of research has also developed connections of misconceptions to math anxiety, as well contributions of acceptance of misconceptions about mathematics, mathematical self-concept, and arithmetic skills to mathematics anxiety. In a study of 92 adult students aged 18 to 57 with a median age of 27, (16 males and 76 females) taking a statistics course, results showed that acceptance of misconceptions and mathematical self-concept were significantly related to mathematics anxiety. The combination of misconceptions, mathematical self-concept and arithmetic skills was significantly related to statistics course performance. Older students returning to school after several years’ absence were the ones most debilitated by negative attitudes toward mathematics. It was concluded that mathematics anxiety involves a mechanistic, non conceptual approach to math, a low level of confidence and a tendency to give up easily when answers are not immediately apparent.

As mentioned, misconceptions must be deconstructed, and teachers must help students reconstruct correct conceptions. 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.