, she responded in a
manner that seemed to be applying a rule which says "you can always change
subtraction to division when doing logarithms", or in the form of the rule:
Errors Due to Misperception and the Default-Value Model
Fong-lok Lee & Rex M. Heyworth
Faculty of Education, The Chinese University of Hong Kong
Email: fllee@cuhk.edu.hk
The traditional ways for the explanation of errors have focused on general problem-solving behavior such as repairing of impasses [8,9] and domain-specific inference such as misgeneralization [3]. The possibility that errors might be generated by misperceiving when a student is learning how to solve a problem or when he or she is working on a given problem has been overlooked. This study suggests that errors can be caused by misperception and that these misperceiving processes can be simulated by using a model, called the Default-value Model developed by the authors.
Keywords: Cognitive Modeling, Misperception, Errors
1. Introduction2. Errors and Perception of problems
A problem has to be perceived or encoded before it can be solved [6,7,5,2]. External display has to be done through an encoding process so that it can affect the problem solving process. It is therefore possible that errors are caused by how a problem is encoded: when a problem is perceived as unsolvable, an impasse occurs; on the other hand, if a problem is perceived as solvable, there will be no impasse. In the first case, errors may be caused by impasse repairing; in the second case, errors may also occur if the problem is incorrectly perceived. Such errors are referred to as the errors due to misperception.
2.1 Errors due to Misperception
A case found in our study showed that some errors would better be
explained as the student misperceiving the problem situation. When a student was asked to
simplify an expression log 60 - log 6 where the correct response should be , she responded in a
manner that seemed to be applying a rule which says "you can always change
subtraction to division when doing logarithms", or in the form of the rule:
(*) Expression_in_log1 - Expression_in_log2 
where Expression_in_log1 or Expression_in_log2 are expressions involving logarithms. A reason given by the student was that she could always simplify an expression this way, which indicates that the error was caused by applying a rule. A later interview with the same student showed that this is a rule she used successfully in many cases. We therefore cannot say that this student has experienced any impasse before or when solving that problem.
The error also cannot be explained as the result of misgeneralization from the correct rule if misgeneralization is considered as the relaxation of constraints on any part of the correct rule [3]. In this case, the correct rule concerned is:
(**) log a – log b = log a/b
which differs from the incorrect rule in the right-hand side. It seems
unlikely that 
can be
generalized from 
.
It is quite clear that this error was caused by an incorrect rule. Exactly how this rule was learned is not yet known, but the following paragraph tries to explain it in terms of a new model, called the Default-value model.
2.2 Frame System and Default Value
A frame is a structure like a database record with slots and fillers corresponding to fields and values [1]. In each frame, there are slots, each of which represents a certain attribute of the object represented. Also, frames can be joined together by links to form frame systems. A frame can inherit values of slots from another linked frame called the parent frame. A default value means that unless there is a contradictory value assigned to the same slot, the slot will take the default value. The use of default values allows the system to be more economical in terms of memory used. There is evidence showing that we do, at least sometimes, use default values [4].
2.3 Incomplete Learning
Frames are effective means for the storing of knowledge. For example, the rule:
can be
represented as follows:
In learning this rule, it is necessary for the student to notice the differences between the input and the output patterns. These include:
However, when students are learning this rule, they may overlook some of these features. For example, a student may just focus on the peculiar part of the rule stating that the subtraction in the input pattern has to be changed to division, but neglect all the other features. In this case, what goes into the student's working memory might be what is represented by the frame system shown below:
The frame representing the output pattern contains several empty slots, which is the result of incomplete learning. All slots but one are now empty. There could be two ways that this system is stored into long-term memory: either this frame system is stored with the empty slots, or the system is stored with the empty slots filled with default values. In the first case, the learned rule would later become the source of impasse, which then needs to be repaired. In the second case, the most probable default values would be those inherited from the input pattern since these values are at that time present in the working memory. The following shows what is stored in the long-term memory when this happens:
2.4 Misperceiving when Learning
Notice that the example above now shows exactly the error represented by the rule:
.
and this error is formed when a rule is learned, incompletely but with empty slots filled with default values.
2.5 Misperceiving when solving
While a misperceived rule may be stored and become a source of later errors, an error may also be generated when a student misperceives the given problem. One example in our study serves to illustrate this:
During the process of simplifying a logarithmic expression, a Secondary
4 (Grade 9) student came across a situation where the expression “0.4771 - 4.771"
was to be simplified. This is a problem that every secondary four student should be able
to solve easily. However, the student expressed it as 
. The student was actually using a rule:
(***) 
Expression1 - Expression2
to calculate the answer. An interview showed that this student seemed to think that since the expression to be simplified was about logarithms and that she was applying the rule (****)shown below to solve the problem:
(****) 
where Expression_in_log1 and Expression_in_log2 are two expressions involving logarithmic functions.
Exactly what made the student think in this way needs further investigation, but it is quite clear that the error was caused by using a rule in a misperceived situation, although the rule applied might also be wrong in that particular situation.
2.6 Default-Value Model for perceiving a problem
Errors due to misperception during the problem solving process can also be simulated by using the Default-Value Model:
Three frames are involved in this problem solving process. The first
one represents what the student observed with the slots Term 1 and Term 2 missing. This
means that the student did not notice what kind of expressions these were. The second one
represents what was perceived by the student with the empty slots filled with what the
student thought they should be. The student then solved the problem by using a previously
learned rule: 
which is represented in the diagram as the process from the second frame to the third
frame.
3. Conclusion
This paper argues that besides impasse-repairing and misgeneralization, misperceiving can also be a cause of errors either when a student is learning a rule or when the student is solving a problem. Both situations can be simulated by a model called the Default-value model.
The basic argument of the Default-value Model is that it models a rule with a frame system and that errors are explained as being caused by filling the empty slots of the frames with default values. Slots are empty because of misperceiving. Default values are inherited from the parent frame, in this case, the input frame.
Since errors are caused by inheriting default values, and this inheriting process may not be rational, this might suggest that errors may not be the result of rational thinking processes. Further investigation is needed to prove this.
Default-value Model also has the advantage that its constituents, i.e., frames, can be easily incorporated into any intelligent tutoring system. Such a system may shed more light on the causes of students’ errors in the future.
Reference
