Research in Brief - January 2008 - Volume 108 (1)

Research on Students' Misconceptions to Improve Teaching and Learning in School Mathematics and Science

Xiaobao Li and Yeping Li
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Mathematics and science are important for success in school, but learning mathematics and science can be a difficult task to some students. Efforts to improve students' learning of school mathematics and science have led to reform efforts in curriculum and instruction over the past decades (e.g., National Council of Teachers of Mathematics [NCTM], 2000; National Research Council [NRC], 1996). Yet, “In the 2000 National Assessment of Educational Progress, only 17 per cent of grade 12 students nationally performed above a basic level of competence.” (Ball, 2003, p. xi) As the improvement of mathematics and science learning is an extremely complex activity that requires coordinated efforts from multiple resources, good research support is very important. In particular, research on students' learning difficulties should help lay a foundation for developing effective interventions, and thus hold much promise for the improvement of teaching and learning school mathematics and science.

Students' learning difficulties can be presented in the form of errors. But not all the errors that students make are the same. Some errors in procedures can simply be due to students' carelessness or overloading working memory (Lemaire, Abdi, & Fayol, 1996). Some errors in procedures can be caused by faulty algorithms or “buggy algorithms” (e.g., Brown & Burton, 1978). Other errors can have certain conceptual basis and can be termed as “misconceptions”. What follows is an example used to illustrate the subtle differences among errors, buggy algorithms, and misconceptions. Falkner, Levi, and Carpenter found that many sixth-grade students incorrectly filled the box in “8 + 4 = []+ 5” with 12 or 17. Such wrong answers clearly indicate that children had a partial understanding of the equality and equal sign (Falkner, Levi, & Carpenter, 1999). According to the above explanations of errors, bugs, and misconceptions, 12 or 17 can be thought of as an error. Since the correct algorithm for this problem normally involves the sum of 8 and 4, then subtracting 5 from 12; a faulty algorithm therefore can be that students only conducted the first step and got the definite value “12.” Another faulty algorithm in this problem was to add all the numbers and got “17.” Underlying all these faulty algorithms may be students' misconceptions of the equal sign, that is, interpreting “=” as “to do something.” Another probable misconception is that students only understood “8 + 4” as a computation process without understanding “8 + 4” as an expression that can also be used to represent certain amount and an object of a higher level of mathematical thinking.

There are many more studies on students' misconceptions in school science than mathematics. As surveyed by Duit (2007), there are over 8000 studies in science education literature that reported the existence of misconceptions or alternative conceptions. Efforts to understand why some systematic errors are so robust to change over time have led researchers to examine the conceptual basis of students' errors. In the research on students' robust misconceptions of science concepts, Chi and her colleagues (e.g., Chi, 2005; Chi & Roscoe, 2002) developed a theory to explain why some science misconceptions are robust. They indicated that student's ontological knowledge and the actual ontological categories may not correspond. Many robust misconceptions are caused by a mismatch between students' conception and the reality at the ontological level. In other words, “robust misconceptions are mis-categorizations across ontological boundaries in that a member of one ontological category is misrepresented as a member of another ontological category.” (Chi, 2005, p. 164) For example, in learning the science concept of current, students initially tend to think current as a material substance which is something like water (Slotta, Chi, & Joram, 1995). The actual category of current is a process of interaction. As a result, there is an ontological difference between students' initial conception of current and its actual attributes. Thus, it is often difficult to change students' misconception of current during formal schooling. In the same article, these researchers also developed a method to examine whether students understand a concept as a process or an object. Their rationale was that “if novices have classified a concept as a material substance, their explanations should contain verbal predicates that correspond to the ontological attribute of that category.” (Slotta et al, 1995, p.378) They coded the predicated words in students' explanations to some well-designed questions. In a recent article about students' misconceptions, Slotta and Chi (2006) employed the same method and gave an example to show how this method works:

For example, if a subject said, “The current comes down the wire and gets used up by the first bulb, so very little of it makes its way to the second bulb, then these four (underlined) predicates were taken as evidence that subjects conceptualized current as a substance-like entity with attribute of (1) “moving,” (2) “can be consumed,” (3) “can be quantified,” and (4) “moves.” (p. 265)

In contrast, recent studies focusing on errors and misconceptions in school mathematics are difficult to find (Barcellos, 2005), although some past studies were carried out to focus on students' errors caused by correctly using buggy algorithms or incorrectly selecting algorithms in elementary arithmetic (Brown & Burton, 1978; Brown & VanLehn, 1982) and in elementary algebra (Matz, 1982; Sleeman, 1982). In the efforts to explore the conceptual basis of students' procedural errors or buggy algorithms, existing research tended to attribute students' learning difficulties to their underdevelopment of logical thinking (e.g., Piaget, 1970), lack of understanding of mathematical principles underlying procedures (e.g., Resnick et al., 1989), or poor understanding of mathematics symbols (e.g., Wearne & Hiebert, 1989). Rather than characterizing students' difficulties or misconceptions in terms of a deficiency model, McNeil and Alibali (2005) argued that students' misconceptions may be caused by their previous learning experiences. Nevertheless, the above researchers mainly focused on students in understanding their difficulties or misconceptions instead of mathematics concepts themselves. Few studies have focused on the internal structures of some mathematics concepts that caused the misconceptions for most students. In fact, students' misconceptions of some mathematics concepts are particularly robust to change even under reasonably “good” teaching over a long time. For example, the use of the “equal sign” is a basic topic in elementary mathematics, however, students' understanding of “=” as “to do something” at early grades caused students' misconceptions related to “equation” in the middle school level (e.g. Carpenter, Franke, & Levi, 2003), and even for high school and college students who often have trouble understanding and using the equal sign (Barcellos, 2005; Clement, Lockhead, & Monk, 1981). Thus, it is very important for mathematics educators to examine why some concepts in school mathematics are so difficult for students to learn.

Given the numerous research on robust misconceptions in school science, it is natural to think whether and how we can adapt some theories and research methods of robust misconceptions in sciences to school mathematics (e.g., Vamvakoussi & Vosniadou, 2004). The content difference between school mathematics and science may let us perceive easily their differences, but not connections, between misconceptions in school mathematics and science. However, the theory developed by Sfard (1991) to account for students' learning difficulties in school mathematics shed some light to possible connections with misconceptions in school science. In particular, Sfard analyzed mathematical concepts in two fundamental ways: structural and operational, which respectively results in two separate entities - “objects” and “process” - similar constructs as those used by Chi and her colleagues (e.g., Chi, 2005; Slotta & Chi, 2006). As an example, Sfard indicated that there are two stages in Children's learning development of “number.” When children studied the concept of “number,” they started from “counting.” This was the stage of “process,” which was natural and relatively easy for them.

However, children at last needed to convert the counting process to an abstract concept of “number.” This was the stage of “object,” as Sfard and Linchevski (1994) argued that students need to transit from process-like thinking to object-like thinking in order to understand a concept completely. Possible connections between Sfard's and Chi's theories present the potential to &lquo;borrow&rquo; research on students' misconceptions in school science to develop a theory to explain why some misconceptions in school mathematics are robust to change (Li, 2006). Certainly, it is still important to keep in mind the differences between mathematics concepts and science concepts. For example, almost all mathematics concepts in numbers and algebra were constructed by mathematicians and thus have no concrete representations in daily life but abstract. In contrast, almost all school science concepts, such as heat, stem from students' lives. The differences between school science and mathematics concepts should be noted when we try to develop a model to explain students' misconceptions in school mathematics.

Existing research on students' errors and their conceptual sources have laid a foundation for developing possible interventions on some concepts, which is not only critical to learning (McNeil & Alibali, 2005) but also to teaching mathematics and science effectively. Resnick (1982) attributed students' learning difficulties to conceptual learning: “difficulties in learning are often a result of failure to understand the concepts on which procedures are based.” (p. 136) Further research on students' misconceptions in school mathematics and science should empower teachers to use proper strategies to help students, because “one of the greatest talents of teachers is their ability to synthesize an accurate ‘picture,’ or model, of a student's misconceptions from the meager evidence inherent in his errors.” (Brown & Burton, 1978, p.155-156)

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