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MSOR Connections 18(1) – journals.gre.ac.uk 37
RESEARCH ARTICLE
Creating a Taxonomy of Mathematical Errors for
Undergraduate Mathematics
Shauna Ford, School of Mathematics, Cardiff University, UK. Email: FordS6@cardiff.ac.uk.
Jonathan Gillard, School of Mathematics, Cardiff University, UK. Email: GillardJW@cardiff.ac.uk.
Mathew Pugh, School of Mathematics, Cardiff University, UK. Email: PughMJ@cardiff.ac.uk.
Abstract
In this paper we develop a taxonomy of errors which undergraduate mathematics students may
make when tackling mathematical problems. We believe that a taxonomy would be useful for staff
in giving feedback to students, and would facilitate students’ higher-level understanding of the types
of errors that they could make.
Keywords: assessment, feedback, errors.
1. Introduction
There has been a considerable amount of research over the last century into mathematical errors
(Radatz, 1980). Typically this research is in the context of learning mathematics in school (e.g.
Radatz, 1980; Matz, 1982; Kieran, et al., 1990; Foster, 2007). Such studies tend to focus therefore
on errors which are either arithmetic or algebraic in nature, such as errors in long division, or
misinterpreting as . An approach which seems to receive particular attention in the
U.S. is error analysis (e.g. Ashlock, 2010; Idris, 2011). In this approach pupils’ errors are
systematically recorded by the teacher, and analysed for patterns so that teacher can then plan what
potential remedial action will be necessary to correct any underlying misconceptions.
Whilst research into errors made by pupils in a school context can be of benefit to teachers in higher
education, the contexts are also very different. School pupils will possess a wide range of
mathematical abilities, and many will have a dislike or even fear of mathematics. On the other hand
mathematics undergraduate students possess a strong mathematical ability and have chosen to
study the subject further. Thus one would hope that many of the errors made by pupils, resulting
from a misunderstanding of even basic concepts within mathematics, would not be made by
mathematics undergraduates. The approach to mathematics also tends to be very different in the
two contexts. In school the focus is almost exclusively on algorithms to solve problems, whilst
undergraduate mathematics will also focus on understanding concepts and proving results.
Therefore the types of error made in higher education will typically be of a different nature to those
in primary or secondary education.
Much of the above research, as well as more general studies on mathematical errors, focuses on
understanding the underlying cognitive causes of these errors, either in order to understand the
cause of specific errors, or more generally to identify the mechanisms underlying these errors. It is
argued that most mathematical errors are causally determined, and very often systematic (Radatz,
1980). Radatz (1979) identified five error categories: (1) errors due to language difficulties, (2) errors
due to difficulties in obtaining spatial information, (3) errors due to deficient mastery of prerequisite
skills, facts, and concepts, (4) errors due to incorrect associations or rigidity of thinking, and (5) errors
due to the application of irrelevant rules or strategies. Ben-Zeev (1998) constructed a taxonomy of
mathematical errors and attempted to identify the causes of these errors by integrating findings from
different studies. The focus in this and other research is to understand why a student makes an error.
For example, a student may over-generalize an algorithm which holds in one context to a structurally
38 MSOR Connections 18(1) – journals.gre.ac.uk
similar context where the algorithm no longer works, something Ben-Zeev calls syntactic induction
(Ben-Zeev, 1998).
It will often however be difficult, if not impossible, to diagnose the underlying error in a student’s
reasoning or understanding solely from the student’s written solution to a problem. Therefore this
paper will not focus on this, but rather on classifying the particular types of errors students make
when attempting to solve mathematical problems. Such a classification should provide enough
details so that a student can identify what it is they have done wrong, whilst keeping the number of
classes as small as possible. We believe that creating a taxonomy of errors is useful for the following
reasons:
it will be a useful resource for students to see which errors to avoid, some of which may not
have been appreciated previously;
it could be incorporated into a feedback tool for lecturers to enrich the feedback offered to
students;
it would allow for the consideration of relationships between different types of error.
Whilst this paper implicitly assumes that the students we consider are undergraduate mathematics
students, the developments in this paper could also be applied to GCSE or A-level mathematics
students.
This work was undertaken as part of an undergraduate summer project by the first author. The paper
is structured as follows. In section 2 we consider the definition of error that we use in this paper, and
what causes errors to take place, before relating this information to errors in mathematics. The
taxonomy is given in section 3.
2. Human Error
Human error is a failure of a planned action to achieve a desired outcome. Errors can be made in
one of two ways – either the plan itself may be inadequate, or else the execution of that plan may
include actions that are unintentional and which do not lead to the desired outcome, as illustrated in
figure 1.
Figure 2. Occurrence of Human Errors.
Failures in planning are often referred to as mistakes rather than errors. There are two types of
mistakes: knowledge-based and rule-based (Reason, 1990). Knowledge-based mistakes occur
when an individual has an inability to reach an end goal because of a lack of knowledge. Rule-based
mistakes occur when an individual wrongly modifies an established process. Such mistakes are more
. Plan
1
. Action
2
Desired
Outcome
Adequate
Adequate
Inadequate
Unintentional
Intentional
Intentional
Error
Success
Error
3
. Outcome
MSOR Connections 18(1) – journals.gre.ac.uk 39
likely to go unnoticed when the outcome is not specifically known. The modification is likely to be
informed by previous successful experiences (Rasmussen, 1986, p.102). Rule-based mistakes fall
into two categories:
• Misapplication of a good rule: Occurs when an individual applies a rule which may be
perfectly adequate in another situation, but which may not meet the conditions and demands
of the problem being considered (Reason, 1990, p.75). Such errors are more likely to occur
when an individual has applied the rule successfully for a previous problem.
• Application of a bad rule: A good rule may become bad following changes that an individual
makes that are not thoroughly considered. This may be from the alterations not being
managed appropriately, or the creation of a bad rule from incorrect knowledge. This can
appear on varying levels; the rule could be entirely wrong, the rule may be clumsy or
inefficient but still achieve the desired outcome, or the rule could be inadvisable since whilst
leading to a good approximate solution, repeated use may worsen this approximation.
Unintentional actions are classed as skill-based errors. These often occur when implementing
elementary or standard procedures, due to a lack of consciousness or control (Rasmussen, 1986,
p.100). Skill-based errors fall into two categories: memory lapses and slips of actions.
• Memory lapse: These errors include losing place in a sequence of steps, forgetting to do
something, or forgetting the overall plan entirely.
• Slip of action: An unintentional action that occurs at the point of execution. This error is often
caused by a process being performed subconsciously, skipping or reordering steps in a
procedure, or experiencing a distraction.
The skill-rule-knowledge framework described above only offers a partial account of possible deviant
behaviour (Reason, 1990). Humans plan and execute their actions in social environments that may
affect their performance. Whilst mistakes and skill based errors are defined as errors made in the
individual’s cognitive stages, their behaviour may also be altered by the situation’s social context.
Violations are deliberate alterations considered necessary by the individual to adjust to external
influences (Reason, 1990, p.195). Hence the violator is not always entirely blameworthy for the
decision made. The following three types of violation are distinguished:
• Routine violations: These occur due to natural instinct to take the process that requires the
least amount of effort. This becomes habitual and forms a set pattern of errors in their
behaviour.
• Situational violations: An individual alters their behaviour due to a change in their social
surroundings. These changes can include excessive time pressure, stress, workplace
design, and inadequate or inappropriate equipment.
• Exceptional violations: These occur when an individual adopts a cause of action known to be
usually incorrect but determines that the current situation is an exception.
Violations and errors from the previous skill-rule-knowledge framework can coincide or appear alone.
The classification of human errors described above is illustrated in figure 2.
40 MSOR Connections 18(1) – journals.gre.ac.uk
Figure 2. Human Error Types.
How might these types of errors appear in the context of a student attempting a mathematics
problem? Suppose a student was answering the question:
Differentiate the function .
This requires using the product and chain rules to obtain an answer of . A ‘slip
of action’ might be manifested as a numerical slip-up (such as writing the coefficient of the derivative
as 5 rather than 6, possibly through subconsciously confusing with ), or a careless
error in writing the solution (such as writing by mistake). If the student could not recall the chain
rule then this would be a ‘memory lapse’, whereas if they did not know the chain rule then they would
likely make a ‘knowledge-based’ error. If they had incorrectly recalled the chain rule, then the error
would be an ‘application of a bad rule’. On the other hand, if the student had (wrongly) integrated the
function correctly, they are likely to be guilty of ‘misapplication of a good rule’. A ‘routine violation’
could occur if a student had made the same error often enough so that they no longer realised it was
an error, for example, writing for the derivative of . A ‘situational violation’ might be more
likely if the student had to answer this question in an examination, perhaps due to the stress and
time-pressures of the situation. An ‘exceptional violation’ may occur if a student is presented with the
question ‘Show that the derivative of is given by ’, where
they might ‘violate’ a rule in a desperate attempt to arrive at a solution which matches the given
answer.
3. Creating a Taxonomy of Mathematical Errors
A taxonomy is the “theoretical study of classification, including its bases, principles, procedures and
rules” (Simpson, 1961, p.11). It is a way of classifying entities verifiable by observation (Bailey, 1994,
p.6). A successful taxonomy will provide classes that are both exhaustive (an appropriate class for
each entity) and mutually exclusive (only one suitable class for each entity) (Bailey, 1994, p.3).
Human Error
Skill-based Error
Mistake
Violation
Situational
Exceptional
Routine
Slip of
Action
Memory
Lapse
Knowledge-
based
Rule-based
Misapplication
of a good rule
Application
of a bad rule
