Error Patterns and Misconceptions in Math
Learning is a complex process that involves a number of systematic steps, which determine the success or failure of the process. Three elements: slips, error, and mistakes, are often identified with the process of learning and more so in mathematics. A slip refers to an incorrect response as a result of processing. Slips are often observed among both experts and novices, and are easy to identify and usually instantly corrected. Errors, on the other hand are incorrect responses and answers due to poor planning, errors are very common in the sense that they mostly apply in similar circumstances. Misconceptions are underlying beliefs and principles in the mind and are responsible for error occurrence. The ability to make suitable and reasonable adjustments to promote access and attainment of all students is an important aspect of the teaching process. Therefore, it is important for teachers to be able to identify the learner’s misconceptions that bar them from attaining new conceptual learning. This paper addresses the common errors and misconceptions in mathematics, and effective ways teachers may employ in addressing them.
According to the theory of Constructivism of learning, concepts are not directly picked from experience, but rather an individual’s capacity to learn from previously existing knowledge (Olivier, 1989). It implies that a child interprets new concepts based on pre-existing knowledge from past experiences. Difficulties in relating the new knowledge to the existing knowledge base usually give rise to misconception, which results in occurrence of errors. Research shows that most misconceptions in mathematics at both elementary and secondary levels emanate from overgeneralization and overspecialization. Overgeneralization is the act of rushing into a conclusion without having enough information at hand. For instance a student may take the sum of two numbers to be the number written on the right of an equal sign and would therefore perceive these two equations as sums.
Overspecialization refers to the notion where a student overspecializes during the learning process, for instance, a student may believe that fractions must have equal denominators for them to be summed or subtracted. These misconceptions usually result in a significant amount of errors.
The main error types resulting from mathematical misconceptions are;
- Incomplete Answers; student partially solves a problem
- Misused Data; student follows the correct procedure, but gives an incorrect final conclusion
- Technical/ Computational error; errors associated with manipulation of algebraic symbols
- Distorted Definition; student changes the definition essential for solving a problem
- Error originating from misconceptions of previously learned material; a learner fails in a applying a previously learnt procedure in the same unit.
Awareness of the common mathematical errors and misconceptions among learners is an important tool for teachers to evaluate student thinking and learning procedures (Schnepper & McCoy, 2013).
To help learners overcome these misconceptions and errors, teacher’s approaches ought to focus on procedural and conceptual aspects. Conceptual knowledge refers to knowledge that focuses on relationships, whereas the procedural aspect involves putting emphasis on symbolic representation and algorithms (Chick & Baker, 2005). The theory of behavioral learning states that “The effects of incorrect rules of computation, as exhibited in faulty performance, can most readily be overcome by deliberate teaching of correct rules,”(Olivier, 1989) this implies that re-teaching a misunderstood concept is a measure to help students overcome misconceptions and errors. A second technique a teacher may employ is cognitive conflict; this refers to the act of setting up a situation that allows a learner to identify a contradiction between previous knowledge and new ideas, and thus enable them to re-evaluate their approach. A third technique involves the teacher probing students’ thinking process by asking them to explain their procedures to help them identify their errors.
When examining Fred’s solutions involving multiplication problems, it is evident that a common error pattern occurs throughout the exercise. The student is able to compute the first two problems correctly because they do not involve renaming. However, Fred incorrectly computes all the problems that require renaming. An analysis of Fred’s errors reveals a misconception in a certain multiplication algorithm. The student multiplies the first digits correctly, and adds the value carried forward to the second before multiplication instead of the vice versa. The misconception here is that the student believes that the addition element should always precede multiplication and therefore Fred adds every value carried forward before embarking on the multiplication aspect. Fred’s problem can be remedied by teaching the missing concept and putting an emphasis on the correct algorithm procedure (Olivier, 1989).
“An algorithm is a step-by-step procedure for accomplishing a task, such as solving a problem,” (Ashlock, 2002) it is an essential element in the field of mathematics. It is essential for a teacher to teach algorithms for a number of reasons, principally being power; they apply to a wide variety of problems and accuracy; applied correctly, an algorithm usually produces the correct answer. In Fred’s case, there is a mishap in the algorithm involving multiplication and more specific multiplication that involves “carrying.” As previously stated in this paper, it is essential for teachers to diagnose learners’ misconceptions, and diagnose their error patterns in order to determine effective means of addressing them.
Ashlock, R. B. (2002). Chapter 1 / Computation, Misconceptions, and Error Patterns. In R. B. Ashlock, Error Patterns in Computation: Using Error Patterns to Help Each Student Learn (10th Edition) (p. 302). Upper Saddle River, N.J: Merrill Prentice Hall.
Chick, H. L., & Baker, M. K. (2005). Investigating Teachers’ Responses to Student Misconceptions. Melbourne: University of Melbourne.
Olivier, A. (1989). Handling Pupils’ Misconceptions. Stellenbosch 7600: Department of Didactics, University of Stellenbosch.
Schnepper, L. C., & McCoy, L. P. (2013). Analysis of Misconceptions in High School Mathematics. Networks: Vol. 15.