1
The negative sign and exponential expressions: Unveiling students’
persistent errors and misconceptions
Richard Cangelosia,*, Jo Olsonb, Silvia Madrida, Sandra Coopera, Beverly Hartterc
aDepartment of
Mathematics, Washington State University, United States
bDepartment of
Education, Washington State University, United States
cDepartment of
Mathematics, Oklahoma Wesleyan University, United States
ABSTRACT
The goal of this study was to identify persistent errors that students make when simplifying
exponential expressions and to understand why such errors were being made. College students
enrolled in college algebra, pre-calculus, and first- and second-semester calculus mathematics
courses were asked to simplify exponential expressions on an assessment. Using quantitative and
qualitative methods, we found that an incomplete understanding the concept of negativity was the
source of most of the students’ errors. We conjecture that students must develop a deeper
understanding of additive and multiplicative inverses to develop a more abstract understanding of
negativity.
Keywords: Exponentiation, negative numbers, conception, concept image, additive and
multiplicative inverse additive and multiplicative identities
* Corresponding author.
E-mail address: rcangelosi@wsu.edu (R. Cangelosi)
2
1.
Introduction
Algebra provides the foundation for advanced mathematical thinking; and proficiency in
algebraic manipulations is essential to students who want to enter science, technology, engineering,
and mathematics (STEM) careers (Liston & O'Donoghue, 2010). Research on the development of
algebraic reasoning is an emerging focus area in mathematics education (e.g., Kieran, 2007; Seng,
2010; Vlassis, 2002a, 2002b; Warren, 2003). Most studies focus their attention on functions (e.g.,
Dugdale, 1993; Thompson, 1994; Vinner, 1992) or solving linear equations (e.g., Sfard &
Lincheviski, 1994; Slavit, 1997). Comparatively few studies investigate the simplification of
algebraic expressions (Ayres, 2000; Sakpakornkan & Harries, 2003), a skill which requires students
to use their understanding of variables and to interpret mathematical symbols accurately. In
addition, research on students’ understanding of the negative sign is limited, particularly in the
context of exponential notation (Kieran, 2007).
The present study grew out of a week-long workshop with approximately 40 high school
juniors and seniors. During the workshop, which focused on exponential and logarithmic
expressions and equations, it became obvious that students had a fragile understanding of
exponential expressions. We recognized that the errors made by the high school students were the
same as those frequently committed by university level students. This led us to examine more
closely students’ facility in working with exponential expressions. In this study, we sought to
identify students’ persistent errors made while simplifying exponential expressions and to
understand why students made these errors. We define a persistent error to be an error that
students continue to make as they progress through more advanced courses and which is based on
an under-developed mathematical concept.
2.
Theoretical framework
3
This study used an investigative approach based on a constructivist perspective in which
learners construct their knowledge about mathematical ideas from their own experiences. The
process of learning requires the learner to adapt existing knowledge from previous experiences to
accommodate new ideas. The focus of research from the constructivist framework places primacy
on the individual and how knowledge is constructed. To understand the development of students’
knowledge regarding exponential expressions, we use a framework proposed by Sfard (1991,
1992) which builds upon the notions of concept image and concept definition (Tall & Vinner, 1981).
A mathematical concept is a complex web of ideas developed from mathematical definitions
and mental constructs (Tall & Vinner, 1981; Vinner, 1992; Sfard, 1991, 1992). Tall and Vinner
described these two components as concept definition and concept image. They use the term
concept image to “… describe the total cognitive structure that is associated with the concept, which
includes all the mental pictures and associated properties and processes. It is built up over years
through experiences of all kinds, changing as the individual meets new stimuli and matures.” (p.
152). A concept definition is a set of words that is used to specify the concept. The definition may be
phrased in language accepted by the mathematical community, in everyday language taught by
teachers, or in the students’ own words as they understand it. As individuals integrate definitions
and images, the concept image becomes more sophisticated. However, individuals may create
idiosyncratic images and definitions that interfere with the development of the concept or with the
development of new concepts (Vinner, 1992). Sfard uses the term concept to mean a mathematical
idea within “… the formal universe of ideal knowledge,” and the term conception to represent “…
the whole cluster of internal representations and associations evoked by the concept.” Sfard’s use of
conception is similar to Tall & Vinner’s notion of concept image. However, her use of the word
concept, while similar to Tall and Vinner’s use of concept definition, does not include informal
language. We adopt Sfard’s framework because it provides greater detail into the process of
learning mathematics.
4
Sfard (1991, 1992) develops a theoretical model for the learning of mathematical concepts
that encompasses both operational (procedural, algorithmic) understanding and structural
(conceptual, abstract) understanding, characterizing both as necessary and complementary. Based
on historical examples and cognitive theory, she asserts that when learning new mathematics, the
natural entry point is through an operational approach. She claims that the precedence of the
operational aspects of conception over the structural aspects is an invariant characteristic of
learning “… which appear[s] to be quite immune to changes in external stimuli” (1991, p. 17). The
transition from an operational understanding to a structural understanding occurs through stages
and is a long and “inherently difficult” process.
When learning a new concept, a natural starting point is through a definition. According to
Sfard (1991), some mathematical definitions treat concepts as objects that exist and are
components of a larger system. This is considered a structural conceptualization. On the other
hand, concepts can also be defined in terms of processes, algorithms, or actions leading to an
operational conception. A structural conception requires the ability to visualize the mathematical
concept as a “real thing” that exists as part of an abstract mathematical structure, whereas an
operational conception implies more of a potential that requires some action or procedure to be
realized. Sfard emphasizes that these two conceptions are not mutually exclusive; they are
complementary. We are dealing with duality, not dichotomy. The operational and structural
aspects of conception can be considered as two sides of the same coin; both are critical to building a
deep understanding of mathematics.
As students move from an operational to a structural understanding, they go through three
stages: interiorization, condensation, and reification. During the first stage, interiorization, the
student becomes skilled at performing processes involving the concept until these processes can be
carried out mentally and with ease. For example, an individual may start with the concept of
5
inverse operations as “the opposite of addition is subtraction” and “the opposite of multiplication is
division.” At this stage students can use this concept to solve basic linear equations. However, they
may not recognize the role of the additive and multiplicative identities in the process.
During the second stage, condensation, the learner is able to think about a complicated
process as a whole without needing to carry out the details. The person is able to break the process
into manageable units without losing sight of the whole. In this stage, there is also a growing
facility with moving between different representations, recognizing similarities, and making
connections. This stage lasts as long as the mathematical notion remains tied to certain processes.
For example, students may recognize zero as the additive identity, one as the multiplicative
identity, and the role of the identity elements. By recognizing the similarities between additive and
multiplicative inverses, students begin to see both as specific examples of the concept of inverse.
A concept is reified when the student can perceive the concept as an object and use it as an
input to create more advanced ideas (Sfard, 1991, 1992). Reification represents a significant shift
in thinking, one in which the concept is suddenly seen as part of a larger mathematical structure. It
is at this stage that students begin to operate with a concept as an object and as the input into new
processes. In fact, reification frequently requires being exposed to more advanced concepts which
require this new object as a building block. For example, the concept of inverse applied to number
may be reified when students need to extend the concept of inverse to functions. The stage of
reification is the most difficult and often happens as a flash of insight.
3.
Previous research
The internet abounds with sites that list common errors students make when simplifying
exponential expressions (e.g., Barnes, 2006; Chiu, Ibello, Kastner, & Wooldrige, 2009; Indiogine,
2008). They relied on anecdotal evidence and did not provide an explanation of why students make
these errors.
6
In an attempt to develop a theory of algebraic computational competence, Matz (1980)
discussed conceptual changes that occur in the transition from arithmetic to algebra that can inhibit
students’ computational competence. Two conceptual changes were emphasized: the use of
notation and the equal sign. She identified several notational conventions that may lead to
difficulties. They include (a) the dual usage of the plus and minus signs as both binary and unary
operators, (b) the concatenation. Concatenations with numbers and letters are used to denote
place-value notation, multiplication in algebra, and orders of operations. With respect to the equal
sign, Matz noted that algebra students often confuse tautologies and conditional statements. For
example, students might attempt to solve a tautology by setting it equal to zero or they might
misinterpret a conditional statement as a tautology and try to figure out how one side of the
equation has been transformed into the other.
Barcellos (2005) identified certain persistent errors made by postsecondary algebra
students that involved the misuse of the equal sign and the distributive law, and invalid
cancellations when simplifying expressions. When asked to solve the equation, 2𝑥 − 3 = 5, a
student might write 2𝑥 − 3 = 5 + 3 = 8 = 4. Barcellos points out that even though the equal sign is
not used correctly, students can often follow their own reasoning and arrive at a correct answer. He
refers to this as “notational abbreviation” (p. 82). He categorized errors related to the distributive
law as either invalid or incomplete distribution and concluded that they were generally due to a
careless error rather than an underlying misconception. When erroneously cancelling terms when
simplifying expressions, Barcellos conjectured that students fail to generalize arithmetic rules
learned for rational numbers to irrational or complex numbers.
Research on the interpretation of the negative sign and students’ knowledge of exponents
has focused on middle school through college classrooms (e.g, Chalouh & Herscovics, 1983; Lee &
Messner, 2000; Pitta-Pantazi, Christou, & Zachariades 2007; Sastre & Mullet, 1998; Vlassis, 2002a,
7
2002b, 2004; Weber, 2002a, 2002b). This research body has two central themes: interpretation of
symbols and estimation of magnitudes of exponential expressions. These are summarized below
followed by a discussion of an emergent framework that describes students’ exponential thinking.
3.1.
Interpretation of symbols
The terminology and rules of algebra offer little meaning to many students; they appear
arbitrary (Demby, 1997; Kieran, 2007), similar to the rules of a game. Algebraic rules are
memorized with little or no conceptual understanding and many students have difficulty keeping
track of and applying the rules appropriately. Carraher and Schliemann (2007) described the
difficulties students have in bridging arithmetic to algebra and, in particular, interpreting
mathematical symbols. Kieran (2007) expanded those notions by further discussing the
development of algebraic thinking in middle and high school. She noted that considerable research
exists that describes the ways in which students work with variables, expressions, and equations.
Vlassis (2002a, 2004) examined how middle school students interpreted negativity and
found that eighth-grade students conceptualized negativity as a process linked to the binary
operation of subtraction. Negative nine was easy to interpret in an expression such as 𝑛 – 9 but – 9
alone was more problematic. She concluded that the different uses of the negative sign are
counterintuitive and an obstacle for students.
Students must overcome numerous obstacles to become fluent in algebra, including the
interpretation of operations implied by the positioning of symbols next to each other (Lee &
Messner, 2000). Research on concatenations (Chalouh & Herscovics, 1983) indicated that many
students have difficulty interpreting mixed numbers in which addition is implied, or algebraic
expressions of the form 𝑎 ∙ 𝑏, where multiplication is implied. After many years of high school, some
2
college students misapplied multiplication to mixed numbers. They simplified 3 3 as 3 ∙
2
3
= 2.
8
2
Clearly, these students misinterpreted the meaning of 3 3 by misapplying their understanding of 𝑎 ∙
𝑏. Students also misapply their understanding of 𝑎 ∙ 𝑏 when it takes the form −𝑎2 . In this case, they
misinterpret −𝑎2 to be a positive number.
3.2.
Estimation of an exponential function’s magnitude
Much of students’ work in early algebra is dominated by linear functions. The transition to
exponential functions requires students to conceptualize magnitude in new ways (Kieran, 2007).
Estimating the magnitude of an exponential function is difficult for students partly because they
cannot apply the same reasoning as in the linear case (Mullet & Cheminat,1995; Sastre & Mullet,
1998).
Pitta-Pantazi et al. (2007) used comparisons of exponential expressions as the basis to
propose a model for understanding students’ conceptual development of exponential reasoning.
Without the aid of a calculator, students compared pairs of exponential expressions by choosing the
appropriate relational symbol (>, <, =). These exponential expressions contained numbers which
were too large to calculate using pencil and paper; instead students had to rely on properties of
exponents and their knowledge of number systems.
3.3.
Frameworks to describe students’ exponential thinking
Weber (2002a, 2002b) and Pitta-Pantazi, et al. (2007) examined students’ conceptions
about exponential expressions. Weber looked at post-secondary students’ thinking regarding such
expressions in the context of APOS theory: an action; a process; and then a mathematical object that
is the result of a process. As an action, positive integer exponents represent repeated multiplication
(for example, 𝑛3 = 𝑛 × 𝑛 × 𝑛). As a process, students can imagine the result of exponentiation
without actually performing it, 𝑛3 ∙ 𝑛2 = 𝑛3+2 = 𝑛5. As the result of a process, exponential
expressions are viewed both as a prompt to compute and as a mathematical object that can be
9
manipulated. In this stage of generalization, students can move beyond natural numbers as
exponents to negative numbers and subsequently, rational numbers as exponents.
Further developing these descriptions, Pitta-Pantazi, et al. (2007) identified characteristics
of three levels of students’ understanding of exponents starting with the prototype of 𝑎 𝑥 where 𝑎
and 𝑥 are positive integers. At Level 1, students use the prototype as repeated multiplication and
extend it to positive rational bases. At Level 2, students extend the prototype to include positive or
negative rational numbers as bases and integer exponents but understand 𝑎 𝑥 𝑎 𝑦 = 𝑎 𝑥+𝑦 only when
𝑎, 𝑥 and 𝑦 are positive integers. At Level 3, students extend the prototype to include rational
exponents.
4.
Methods
The purpose of this study was two-fold. First, we sought to identify persistent errors that
students make when working with exponential expressions. Second, we sought to understand why
students make these particular errors. To accomplish these goals, we first administered an
assessment to college students enrolled in four courses: College Algebra, Pre-calculus, Calculus 1
and Calculus 2. Students’ responses were scored, coded and analyzed using quantitative methods to
identify test items that indicated persistent errors. We then used qualitative methods including
semi-structured interviews and conceptual matrices (Miles & Huberman, 1994) to identify
persistent errors and to gain insight into why students were making the errors.
4.1.
Participants
This study was situated at two universities, one larger (more than 20,000 students) and one
smaller (1,000 students). Both universities are seeking ways to improve student achievement in
entry-level mathematics courses and to increase the numbers of students who are able to continue
in science, technology, engineering, and math (STEM) related careers. The course sequence at both
universities follows one of two tracks:
10

College Algebra, Trigonometry, Calculus 1, Calculus 2, or

Pre-calculus, Calculus 1, Calculus 2.
The difference between college algebra students and pre-calculus students is that the
former satisfy a lower prerequisite and take a two course sequence, College Algebra followed by
Trigonometry, to satisfy the prerequisite for Calculus I. College algebra students begin with a more
comprehensive review of prerequisite material and typically have weaker algebra skills then their
pre-calculus counterparts.
Nine-hundred and four (904) predominately freshman and sophomore undergraduate
students enrolled in college algebra, pre-calculus, and first-semester and second-semester calculus
completed a written assessment containing 18 questions about exponential expressions (see
Appendix A). Approximately one-half of the students who completed the assessment were
randomly selected for this study. Data includes 128 assessments from college algebra, 100 from
pre-calculus, 100 from first-semester calculus and 126 from second-semester calculus.
4.2.
Data collection and analysis
Two sets of data were collected to answer our research questions: (1) responses to a
written assessment on simplifying exponential expressions, and (2) student interviews. First, to
determine indicators of persistent errors, students in College Algebra through Calculus 2 completed
an assessment during the first week of the semester and prior to any in-class instruction or review
of properties of exponents. The assessment contained three sections. In Section A, students
simplified eight exponential expressions; in Section B, students compared the relative magnitudes
of six pairs of exponential expressions using the relational symbols (<, > or =); and in Section C,
students determined whether an exponential expression was positive or negative. The questions
for the assessment were drawn from two sources. The questions written for Section A were based
on errors that students made in courses taught by the authors and those written for Section B and C
11
were influenced by Pitta-Pantazi et al. (2007) and Weber (2002a). Students’ responses for each
question were coded as correct or incorrect and statistics were used to identify commonly missed
questions. These questions were used as indicators of persistent errors.
Second, to identify the persistent errors and to understand why student make these errors,
students were selected and interviewed individually. To select interviewees, student scores within
each course were sorted and organized using quartiles. A total of forty students from the four
courses who placed among the top 25% and lowest 25% were randomly selected for an interview.
Four of these students completed the interview. Due to the lack of response from the original
invitations, researchers teaching these introductory courses asked students in their classes from
the aforementioned groups to volunteer for an interview. An additional fourteen students
completed the interview. Semi-structured interviews took place between one and two months after
the written assessment and lasted between 20 to 30 minutes. Students were asked to rework and
discuss five problems that were identified as indicators of persistent errors. Detailed field notes
were taken and student work was collected. Audio tapes of the interviews were made when
possible. Data was collapsed using conceptual cross-case matrix analysis (Miles & Huberman,
1994) and analyzed to characterize student explanations.
5.
Results
To identify problems that may indicate a misconception, quantitative analysis was used.
First, we recount student performance on the written assessment and our process for identifying
problems that indicated a persistent error. Second, we investigate the persistent errors through
analysis of student responses obtained during individual interviews.
5.1.
Results of written assessment
12
Results of the written assessment are presented Tables 1a, b. An examination of the
percentage of students who correctly answered the problems across the four courses indicated a
gradual increase in correct responses with the largest gain typically occurring between Pre-calculus
and Calculus I. As expected, the more advanced students tended to outperform the less advanced
students.
Table 1
Percentage of students who responded correctly to questions on the assessment.
Table 1.a
Percent Correct by
Percent Correct (Section A Questions)
Overall
Section
Total No. Score
Students (%) Section Section Section
1
2 3 4 5 6 7
8
A
B
C
College Algebra
128
45
26
67
44
55
27 49 14 19
8
17
20
Pre-calculus
100
51
31
74
54
62
29 52 12 33
8
31
24
Calculus I
100
68
54
82
75
82
71 63 27 54 25 55
51
Calculus II
126
73
60
85
75
74
69 76 29 68 39 71
58
Table 1.b
Percent Correct (Section B Questions) Percent Correct (Section C Questions)
1
2
3
4
5
6
1
2(a)
2(b)
2(c)
2(d)
College Algebra
96
27
69
69
62
80
NA
77
58
40
31
Pre-calculus
98
36
76
69
74
88
NA
79
72
56
45
Calculus I
98
53
81
81
83
93
NA
94
90
79
72
Calculus II
98
56
83
90
89
94
NA
95
91
84
72
In Section A, students were asked to simplify exponential expressions. Students in all four courses
found this to be the most challenging of the three sections. Performance on questions 1, 3, 5 and 7,
as a whole, steadily improved from college algebra to second-semester calculus. For example, just
13
17% of students in College Algebra correctly answered question A7, which asked students to
simplify
40 − 4−1 .
In contrast, 71% of the students in Calculus 2 answered it correctly. This is the improvement one
would expect to see as students progress and does not represent an example of a persistent error.
In Section B, students compared expressions containing exponents using the relational symbols (<,
> or =). Students were relatively successful in this section with the exception of question B2, in
which they compared −178 and (−17)8. The most common error was to state that these two
expressions were equal. In Section C, students indicated whether an expression was positive or
negative. The students were successful in this section except for question C2.d where a relatively
large number of students responded that 2−3 was negative.
5.2.
Indicators of persistent errors
Our first attempt to identify persistent errors was through 𝑧-scores (see Appendix B). Z-
scores were used to determine whether the proportion of students who correctly answered each
question was significantly different between College Algebra and Pre-calculus than those in
Calculus 2.
Although 𝑧-scores indicate that students made statistically significant progress as they
advanced through the grade levels, it failed to provide a workable way of identifying indicators of
persistent errors. For example, only 8% of College Algebra and Pre-calculus students correctly
answered question A6. Results improved to 39% of Calculus 2 students answering this question
correctly, a statistically significant difference. However, this test does not capture the fact that this
percentage is quite low for students at this level, pointing to the need for a different method of
identifying indicators of persistent error.
14
We then turned to percentile ranks. We first rank-ordered College Algebra and Pre-calculus
students’ test scores and identified those problems at or below the first quartile for each of the
three sections of the assessment. Questions A4, A6, B2, and C2.d were in this grouping for both
College Algebra and Pre-Calculus (see Table 1). We repeated this process for Calculus 2 student test
scores and the same problems emerged. Thus, for this study we consider these problems as
indicators of persistent errors that could be due to possible misconceptions.
5.3
Investigation of the persistent errors
Students were asked to read and re-solve each of five problems (see Table 3), and explain
their work. Four of the five problems were those identified as indicators of persistent errors: A4,
A6, B2, C2d. We also included problem A2 due to its similarity to A4 and A6. Table 3 links the
problems to the knowledge assessed and the levels of students’ conceptual development from PittaPantazi, et al (2007) summarizes student responses organized by question.
Table 3
Problems discussed during student interviews.
Section
Problem
Level of Conceptual
Assessed Knowledge
Understanding
A2
(−8)2/3
3
Simplify a negative base with a rational
exponent
A4
−93/2
3
Simplify an additive inverse with a
rational exponent
A6
(−4)3/2
3
Simplify a negative number with a rational
exponent
B2
−178 <, >,
= (−17)8
1
Compare two numbers raised to a power
15
C 2d
2
2−3
Simplify a number raised to a negative
power and determine whether the result
was positive or negative
5.3.1.
Question A2
This question asked students to simplify the expression (−8)2/3. During the interview,
fourteen out of eighteen students simplified this expression correctly. Of those who were
unsuccessful, one had no idea on how to get started, one dropped the parentheses, and two moved
the negative sign to the exponent.
Students who correctly simplified the expression typically interpreted it as, “Square
negative eight and then find the cube root.” They translated the expression into a procedure which
could lead to the correct simplification. Two others students who correctly simplified the
expression realized that they could first square −8 then compute the cube root or take the cube
root first then square the result.
One student who arrived at the correct answer using incorrect notation wrote (−8)2/3 =
3
3
√−82 = √64 = 4. It appears that the student interpreted −82 as (−8)2 . The other student solved it
3
two ways. The first approach is carried out correctly as (−8)2 = 64; √64 = 4. The student’s second
approach contains the same notational error as above, that is, interpreting −22 as (−2)2 . The
3
2
student wrote, (√−8) = −22 = 4.
All of the students who incorrectly simplified the expression moved the negative sign
inappropriately. One student wrote, (−8)2/3 = −82/3 = −(82/3 ) = −4. When asked why he moved
the negative sign outside the parentheses, he responded, “You don’t need them, they are more for
16
clarity.” Two other students moved the negative sign to the exponent and one wrote (−8)2/3 =
1
8−2/3 = 4. The student explained, “The cube root of eight is two and two squared is four. The
negative sign means that the answer is one over four.” A second student also misinterpreted the
12/3
negative sign signaled that a reciprocal is involved. This student wrote (−8)2/3 = 8
and was unable to further simplify the problem.
5.3.2.
Question A4
This question asked students to simplify the expression −93/2. Thirteen out of eighteen
students simplified this question incorrectly with two later correcting themselves. They read the
problem aloud as, “Negative nine to the three halves power”. These students appeared to include
the negative sign as part of the base. For example, several students rewrote the problem as √−9
3
and remarked, “You can’t take the square root of a negative number.” Another student wrote
(−9) × (−9) × (−9) = 729 and concluded that −93/2 was equal to √−729 = 𝑖√729. When asked
about the difference between question A2 and A4 the student replied, “The first one you square
then cube in this one you cube then square. Parentheses do not affect the answer.” One student
interpreted −93/2 as “ … one over nine to the three halves…” The predominant error was to
interpret the negative sign as part of the base.
5.3.3.
Question A6
This question asked students to simplify the expression (−4)3/2. Eleven out of fourteen
students simplified this question correctly (four students did not discuss their methods of
simplifying the expression with the interviewers). The students who simplified the expression
correctly typically recognized that the square root of a negative number is complex. Of those
17
students that did not simplify the expression correctly, one made a simple computational error.
Another student self-corrected himself. He said and wrote, “It’s (−4)3 = 64 and √64 = 8 so −8. Oh,
can’t have √−64, can’t do this, you get an 𝑖.” Although he used incorrect notation, the student was
apparently keeping track of the negative sign mentally. The third student attempted to simplify the
expression by pulling the negative sign out of the parentheses and in essence concluded that
(−4)3/2 = −(43/2 ) = −23 .
5.3.4.
Question B2
This question asked students to compare the two expressions using the relational symbols
(<, =, or >). Eight of the eighteen students erroneously believed that the expressions were equal
with one student later correcting himself. It is interesting to note that four of the students who
incorrectly interpreted the negative sign as part of the base in question A4, recognized the
distinction here. For example, one student explained, “I know for the one on the left, you do
seventeen to the eighth power first, [and] then use the negative sign. For the one on the right, the
negative sign is ‘involved’ and because eight is even, the result is positive.”
One student misinterpreted the question comparing instead the magnitudes of both
numbers. She asserted, “They are equal because the negative is tagged on outside seventeen to the
eighth and negative seventeen raised to the eight [pointing to (−17)8] has an even power. They are
at the same distance from zero.”
In contrast, five students who gave an incorrect response stated that the parentheses do not
matter. One student explained, “Both mean the same; you don’t really need the parentheses.”
Echoing this sentiment, a calculus 2 student elaborated, “Parentheses do not change anything. The
way that I look at it, parentheses are used to enclose two things [numbers] that are raised to a
power.” When asked to simplify the expressions 15 − 32 and −32 , she said that “15 − 9 = 6 and
18
−32 = 9.” In the former expression, she interpreted −32 as subtract 32 (from 15) and in the latter
expression she interpreted the negative sign as part of the base. She did not recognize the
inconsistency of this interpretation. She explained, “The [negative] sign means to subtract when
there is a number in front of it.”
This question also prompted a few students to reconsider their interpretation of
parentheses. A student initially stated that the two expressions were equal and then reconsidered
his answer when asked whether the parentheses come into play. He responded, “Have to have
parentheses [pause]. Oh, (−17)8 > −178 because with parentheses multiply −17 times −17 an
even number of times.”
5.3.5.
Question C2.d
This question asked students to label the number 2−3 as either positive or negative. During
the interviews students were also asked to simplify the expression. Ten out of eighteen students
simplified the expression incorrectly but only two got a negative answer, 2−3 = −8, and hence
would have provided an incorrect response on the assessment. This suggests that the results
presented in Table 1 may be overly optimistic in terms of revealing the students’ actual
understanding.
Many of the students who erred did recognize that a reciprocal was involved. Five students
simplified the problem and arrived at 21/3. Another student recognized that a reciprocal was
involved. She said, “… gotta flip it over somehow, something goes under to get rid of the negative.”
2
She then wrote 1/3 and said, “It might be [pointing at
2
],
3/1
but I don’t think so.” Two other students
3
simplified 2−3 to 2. One of them corrected himself and reasoned, “Well, two to the minus one is one1
half so two to the minus three is three-halves. [pause] maybe [pause] let’s see.” He wrote 𝑥 −1 = 𝑥;
2
1
1
1
𝑥 −2 = 𝑥, and then stated, “No, it’s [wrote 𝑥 2]. So, [wrote 2−3 = 2𝑥2𝑥2 = 8].” Eight students simplified
19
the expression correctly. Six relied on memorized rules to guide their work while two
demonstrated a deeper understanding by recognizing the connection to multiplicative inverses.
6.
Discussion
We characterize our results by describing two errors associated with the negative sign in
exponential expressions. First, we categorize errors where students inappropriately include the
negative sign as part of base as the sticky sign. For example, students often misinterpreted −93/2 as
equal to (−9)3/2. Here, students interpreted the negative sign as “stuck” to the nine instead of
realizing that −93/2 is the additive inverse of 93/2. Second, we categorize errors where students
either inappropriately move the negative sign or “flip” a number within the expression as the
roaming reciprocal. For example, students interpreted 2−3 as 21/3 or −23 or 2/(1/3) or 3/2. We
conjecture that an under-developed conception of inverse is at the root of these errors. We offer as
contributing factors the effects of language, notation and grouping.
6.1.
The sticky sign
In this subsection we discuss errors associated with the negative sign and the base of an
exponential expression. We examine this error in the context of language, grouping, and notation.
The language and notation in K-12 is at times different from the language and notation used in the
mathematics community. When this is the case, we will refer to K-12 language and notion as school
language and school notation, respectively. For example, the word “opposite” is often used in place
of “additive inverse”, and the notation −𝑎 (note the position of the negative sign) is at times used to
denote the additive inverse of the number 𝑎.
6.1.1.
Language
20
The inherent ambiguity of spoken language has the potential to interfere with the
development of students’ understanding of mathematics (Matz, 1980). In a footnote, Matz claims
that when reading expressions a possible source of confusion is a “lyrical [verbal] similarity” (p.
151). In this study, thirteen out of eighteen students interviewed interpreted −93/2 as (−9)3/2 and
they read both expressions in the same way, “Negative nine to the three-halves,” making no verbal
distinction between the two numbers. Perhaps students read the numbers in the same way because
they believe the numbers are equal. On the other hand, since students hear themselves or others
read the numbers the same way, they may develop the misconception that they are equal.
We ask the reader to pause and to read both numbers aloud in such a way as to verbally
communicate the distinction between the two. Even with a conscious effort to avoid ambiguity,
there is still room for misinterpretation. For example, the number −93/2 might be read, “the
negative of [pause] nine to the three-halves,” or perhaps, “the additive inverse of [pause] nine to the
three-halves power”. While the number (−9)3/2 might be read “the quantity negative nine [pause]
to the three-halves power.” Even with the pauses, it is still difficult to distinguish between the two.
One student noted that even though she read −93/2 and (−9)3/2 in the same way, she knew
that they were different because of how each was entered in the calculator. Her understanding of
the role of parentheses was associated with entering numbers in the calculator. Thus, she
recognized that the negative sign was not “stuck” to the nine in the first expression. The trade off
was that she was not confident in her calculations without the use of a calculator and she was
unable to simplify this expression by hand.
The use of colloquialisms or informal language also has the potential to hinder the
development of a student’s understanding. For example, the term opposite is used extensively in K12 education while the mathematical phrase, additive inverse, is used sparingly, if at all. Students
are exposed to the term opposite in contexts such as, “the opposite of a number is just the number
21
on the opposite side of zero on the number line,” or “the opposite of 𝑎 is – 𝑎” (Coolmath, 2011). If
students are not comfortable with the term additive inverse, they might not fully develop intuition
about negativity. For instance, they might not recognize that – 𝑎 could represent either a positive
or negative number or that −22 is the opposite (additive inverse) of 22 . In addition, limited
exposure to the term additive inverse might interfere with students making connections to other
types of inverses, such as, multiplicative inverse and function inverse.
6.1.2.
Grouping
The most persistent error that we identified in this study was associated with question A.4
(simplify −93/2 ). Only 29% of students in second-semester calculus correctly simplified it. Thirteen
out of eighteen students interviewed initially interpreted −93/2 as (−9)3/2. Evidently, they saw −9
as a signed number in which the negative sign was “stuck” to the base. That is, students view −9 as
a single, inseparable object, that was then raised to the 3/2 power.
One way of interpreting this error is to think of it as a grouping error. Students attach the
negative sign to the number 9, instead of recognizing that the unary operation of negation, or the
operation of taking the additive inverse, is being applied to the number 93/2, not just the 9. One
possible explanation for this is that students often believe parentheses do not matter. This
misconception is reflected in students’ words as we heard many variations of “… parentheses do
not matter” when they were asked to explain their work. Applying this faulty reasoning, two
students wrote
(−8)2/3 = −82/3 = −(82/3 ) = −4 and (−4)3/2 = −(43/2 ) = −23 .
22
Students do not recognize the significance of the grouping indicated by the parentheses. This
indicates that they are working at Sfard’s (1991) interiorization stage where similarities and
differences between the relationships – 𝑛 = (−𝑛) and −𝑛2 ≠ (−𝑛)2 are not yet understood.
When students were asked to compare −178 to (−17)8, several of the students who had
incorrectly interpreted −93/2 as (−9)3/2 were able to distinguish between −178 and (−17)8.
Seven students persisted in believing that the two expressions were equal and five of the seven
explicitly articulated the belief that the parentheses do not matter. One second-semester calculus
student recognized that 15 − 32 = 6, but also claimed that −32 = 9. In the context of 15 − 32 , she
was able to interpret −32 correctly but when −32 was by itself she fell into the sticky sign trap. This
student has clearly not reified the notion of additive inverse. Perhaps if she had a more fluent
interpretation of negation and recognized 15 − 32 as 15 + (−32 ), she might have seen the
inconsistency of her claim. The concept development of negativity involves both subtraction, a
binary operation, and additive inverse, a unary operation. Evidence uncovered from this study lead
us to believe the conception of negativity in the context of subtraction is well developed while in the
context of additive inverse, the concept formation stalled at an earlier stage of development.
Following Sfard’s (1991) model of concept formation applied to subtraction, students first
work with whole numbers whose difference is nonnegative. At the interiorization stage, students
begin to explore subtraction with concrete models such as: the take-away model, the missingaddend model, the comparison model and the number-line model. They move into the
condensation stage when they no longer rely on such physical models and are fluent with
calculations. At this stage they develop an abstract notion of subtraction and become adept at
applying mental algorithms for carrying out subtraction of whole numbers. Reification of
subtraction with whole numbers often occurs once students begin to work with integers. The
23
process of interiorization, condensation and reification repeats itself as the concept of subtraction
is extended to broader number systems and more complex representations of numbers.
The concept of additive inverse (the unary operation of negation) is often introduced
through colored-chip models and number-line models, and the term “opposite” is used in K-12
textbooks and classrooms. Signed numbers are introduced to help students grasp the notion of
negative numbers. Unfortunately, there appear to be limited opportunities for students to move
beyond the interiorization stage. Students become adept at calculations involving negative
numbers but the concept of additive inverse appears to remain elusive. A natural place for this
concept to be developed further is when solving simple linear equations of the form 𝑥 + 𝑎 = 𝑏. An
operational view of solving such equations uses the notion of inverse (opposite) operations. That is,
to undo the action of adding 𝑎, the opposite operation of subtracting 𝑎 from both sides is employed.
This is typically written as
𝑥+𝑎 = 𝑏
−𝑎
−𝑎
.
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝑥
=𝑏−𝑎
A structural view consists of adding the additive inverse to both sides of the equation as illustrated
below.
𝑥 + 𝑎 + (−𝑎) = 𝑏 + (−𝑎)
𝑥+0 = 𝑏−𝑎
𝑥 =𝑏−𝑎
This emphasizes the algebraic structure of real numbers through the use of the additive inverse and
the additive identity. Incorporating this view allows for the condensation of the concept of additive
inverse in the sense that the critical components (additive inverse, additive identity, and their
relationship) are clearly present and transparent. Reification can then occur when students
24
encounter the concept of inverse in other contexts such as multiplicative inverse and functional
inverse.
6.1.3.
Notation
Barcellos (2005) observed that students frequently arrive at a correct answer “… even when
the notation does not conform to mathematical conventions.” (p. 9). In this study, students
occasionally simplified expressions mentally and wrote just enough to remind themselves of what
they were doing. We refer to this as personal shorthand notation. As an example, one student wrote
3
3
(−8)2/3 = √−82 = √64 = 4.
3
While he did not write √(−8)2, the student appeared to recognize that the quantity to be squared
was −8. This personal shorthand notation did not appear to hamper the student’s ability to simplify
the expression; rather it was used to keep track of his mental processes. Barcellos (2005) observed
that students often can follow their own reasoning and arrive at a correct answer. The most
common example of the second problem is to use the equal sign in places where an implication sign
is the appropriate choice. For example, a student may write
1
𝑥
1
1
1
= 6 = 𝑥 = 6 instead of 𝑥 = 6 ⇒ 𝑥 = 6.
The student may be working at Sfard’s (1991) condensation stage where he is performing several
mental calculations to simplify the expression. His personal shorthand notation reflects key steps
in his thought process. It is not clear whether he has made the distinction between −82 and (−8)2 .
Furthermore, the habit of using personal shorthand notation has several disadvantages: (a) it is not
logically consistent, (b) it can lead to more errors when working with complex expressions, and (c)
it interferes with the communication of mathematics.
25
On the other hand, perhaps student difficulties recognizing the difference between −𝑎 𝑥 and
(−𝑎)𝑥 emerges from early instruction on signed numbers, a concept used to facilitate
understanding of negative numbers and subtraction. The notation +9 and −9 is used (note the
position of the positive and negative signs) to help student conceptualize negative numbers as an
object. The sign is to be interpreted as either a point on or a direction along the real number line.
Subtracting −9 from 3 is usually written +3 − −9 (notice that parentheses are not used). In
conventional mathematical notation, the expression is written, 3 − (−9). However, we encounter
college students who use nonconventional notation (e.g., 3 − −9). We suspect that the use of
nonconventional notation stems from their previous work with signed numbers. To identify
instructional practices that may lead to this unconventional notation we examined one popular
middle school mathematics curriculum (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006). In this
curriculum, negative numbers are introduced by signed numbers with a superscript negative sign
(e.g., −𝑎, Figure 1).
Negative numbers are usually written with a dash like a subtraction sign.
−
3 = −3 and −7.5 = −7.5
From now on, we will use this notation to indicate a negative number. This can be confusing
if you don’t read carefully. Parentheses can help. [Emphasis added.]
−
5 − −8 = −5 − −8 = −5 − (−8)
Figure 1. Excerpt from a middle school mathematics curriculum introducing negatives
numbers (Lappan et al., 2006, p. 42).
We suggest that this string of equivalences, −5 − −8 = −5 − −8 = −5 − (−8), leads
to the adoption of unconventional notation. The expression −5 − −8 indicates that the
negative sign preceding the eight is “stuck” to the eight, that is, −8 is an inseparable entity.
It also implies that the parentheses used in conventional notation are not necessary. When
26
−5 − −8 is equated to −5 − (−8), students may conclude that parentheses are not
necessary
6.2.
The roaming reciprocal
In this subsection we discuss errors associated with the negative sign in the exponent of an
exponential expression. We examine this error in the context of language and notation.
6.2.1. Language
As discussed previously, spoken language sometimes interferes with the development of
students’ understanding of mathematics. In this study, many of the students interviewed used the
expression “flipping” instead of the terms reciprocal or multiplicative inverse. When students were
asked what was meant by reciprocal, they responded, “… to flip the numbers”. Students responded
with blank stares when we asked them “What is a multiplicative inverse?”. Students sometimes
figured out that the multiplicative inverse of 𝑎 is 1/𝑎 when reminded that – 𝑎 is the additive inverse
of 𝑎. It appears that the use of the colloquial terms, opposite and “flipping”, hindered students’
understanding of the concept of inverse.
Students that incorrectly simplified the expression 2−3 appeared to have a rudimentary
operational understanding of multiplicative inverse linked to the term “flipping”. The appearance of
the negative sign was a signal for them to form a reciprocal but it was unclear to them what to “flip”
and the reciprocal roamed in its location as the following example illustrates. When rewriting the
expression 2−3 , some students formed the reciprocal of the exponent and wrote 21/3. Another
student formed the reciprocal of the exponent and placed it into the denominator of a rational
2
2
3
number and wrote 1/3. Similarly, two other students wrote 3/1 and 2. In the former case, the
student set the base in the numerator and the exponent in the denominator and in the latter case,
the reverse was done.
27
These students appeared to be working at an operational level where the appearance of the
negative sign was interpreted as a signal to “flip” a number in the expression. Clearly, they failed to
recognize that a negative sign in the exponent represents a multiplicative inverse. With a structural
understanding that links language and the concept of inverse, students might recognize that the
previously discussed expressions could not be the inverse of 2−3 . Sfard (1991) theorized that to
progress within and beyond the interiorization stage, students need to develop both operation and
structural understanding. We theorize that since students have not transferred from colloquial
language (flipping) to the accepted mathematical register (inverse), their opportunity to develop a
structural understanding necessary for condensation and reification is hindered.
6.2.2. Notation
Weber (2002a) explained that the expression 𝑎 𝑥 can be thought of as an action, a process,
and then as a mathematical object that is the result of a process. As an action, students view 23 as
repeated multiplication, 2 × 2 × 2. They do not see 2−3 as an object that can be manipulated nor do
they see 2−3 as the multiplicative inverse of 23 . To develop a structural understanding of the
algebraic properties of numbers, students must link 2−3 with (23 )−1.
In this study, a few students used the definition, 𝑎−1 = 1/𝑎, to remind themselves how to
simplify 2−3 . Looking solely at the notation, they incorrectly generalized the definition to the
erroneous statement, 𝑎−2 = 2/𝑎. They assumed that the base forms the denominator and the
exponent forms the numerator. Applying this incorrect generalization, they wrote 2−3 = 3/2.
Without a structural understanding, their interiorization of the definition, 𝑎−1 = 1/𝑎, failed to
convey the notion of multiplicative inverse. We are unsure whether students’ inability to simplify
2−3 derives from their failure to see 2−3 as an object that can be manipulated or the failure to think
in terms of inverses. Regardless, students need an operational understanding that is linked to
1
reciprocal, 𝑎 𝑥 ∙ 𝑎𝑥 = 1, and an structural understanding that is linked to inverse, 𝑎 𝑥 ∙ 𝑎−𝑥 = 1 in
28
order to move from the interiorization stage into the condensation stage and beyond. Notation
plays an important role in both operational and structural understanding in concept development.
Similar to the development of a structural understanding of addition (Section 6.1.2.), an
opportunity to develop a structural understanding of multiplicative inverse is also available when
solving equations. For example, students typically begin with solving equations of the form 𝑎𝑥 = 𝑏
(𝑎 ≠ 0) by dividing both side by 𝑎. In this operational approach, students simply divide both sides
by 𝑎. A structural approach has the advantage of introducing both the terms and concept of
multiplicative inverse and multiplicative identity into the students’ discourse.
𝑎𝑥 = 𝑏
𝑎 (𝑎𝑥) = 𝑎−1 𝑏
(𝑎−1 𝑎)𝑥 = 𝑎−1 𝑏
1 ∙ 𝑥 = 𝑎−1 𝑏
𝑥 = 𝑎−1 𝑏 = 𝑏/𝑎
−1
In the structural approach, notation is used to link the concept of multiplicative inverse and the
multiplicative identity. We remind the reader that prior to solving equations; students are usually
introduced to or reminded of the associative, commutative and distributive properties of numbers.
We suggest that textbooks and teachers include the concept of multiplicative inverse and
multiplicative identity to complete the discussion of the algebraic structure of numbers.
7.
Conclusions and Implications
The purpose of this study was to identify persistent errors in simplifying exponential
expressions and to gain insight into why such errors are made. We identified two persistent errors,
referred to as the sticky sign and the roaming reciprocal. We propose that both stem from an
underdeveloped conception of inverse, which leads us two implications. First, an underdeveloped
mathematical conception can arrest the development of more sophisticated ideas. Second, language
29
and notation play a critical role in developing students’ conceptual understanding. Following is a
discussion of these implications.
We found that students have an underdeveloped conception of inverse, both additive and
multiplicative. Students simplified binary operations of the form 𝑎 − 𝑏 𝑥 correctly, but they did not
recognize the unary operation −𝑏 𝑥 as the additive inverse of 𝑏 𝑥 . We suspect that student failure to
recognize −𝑏 𝑥 have not transitioned to algebraic thinking. Matz (1980) theorized that dual usage
of the plus and minus signs as both binary and unary operators are necessary for this transition.
When working with an expression of the form 𝑏 −𝑥 , where 𝑥 > 0, students understood that
something needed to be “flipped,” but did not recognize it as the unary operation of the
multiplicative inverse of 𝑏 𝑥 .
One might argue that an underdeveloped conception of inverse is not critical to success in
mathematics, but we argue to the contrary. Students in this study relied on an operational
understanding rather than both an operational and structural understanding, suggested by Sfard
(1991) as essential for reification. In this study, we found that relying solely on inverse operations
instead of the broader notion of inverse, compromised students’ conception of exponential
expressions. We would also expect them to have difficulty solving transcendental equations such as
𝑎 𝑥 = 𝑏 or sin(𝑎𝑥) = 𝑏. To solve such equations students must have at least a procedural
understanding of inverse functions, which may not occur without a solid understanding of additive
and multiplicative inverse.
Knuth, Stephens, McNeil, & Alibali (2006) found that an operational understanding of the
equal sign (a signal to do something) interfered with students’ mathematical development. Middle
school students with a relational understanding of the equal sign were much more successful at
solving algebraic equations. Like this study and Weber (2002a, 2002b), an operational
30
understanding without a structural understanding interferes with students’ mathematical
development.
Language and notation may also hinder students’ development. In particular, school
language and notation may inhibit the development of more sophisticated conceptions that allow
for connections or generalizations to more advanced ideas. When students rely on the term
opposite and “flip” they internalize an action rather than develop a structural understanding that
connects additive and multiplicative inverses. While informal language such as “opposite,” “flip,”
and “undo” help students gain intuition, we suggest that the terms additive inverse, multiplicative
inverse, additive identity and multiplicative identity should also be used regularly in both textbooks
and classroom discourse. Otherwise, it may be difficult for students to recognize that additive and
multiplicative inverses have similar underlying algebraic structures.
Numerous textbooks discuss the algebraic structure of real numbers through associative,
commutative, and distributive properties as a prelude to solving linear equations. Curiously, they
often fail to mention additive and multiplicative inverses and identity elements. Inverses and
identity elements relate to the processes used to solve equations and would complete the
discussion of the field properties of real numbers. The mathematical language used to describe
inverses needs to be connected to standard mathematical notation and concepts. We suggest that
educators be more mindful of the language and notation used in the classroom to help students
make connections between mathematical ideas.
Pitta-Pantazi et al. (2007) researched students’ reasoning when working with exponential
expressions. Based on their results, we anticipated that rational powers would be a stumbling
block for students. However, in our study we found gaps in more elementary concepts, namely
additive and multiplicative inverses, which interfered with our ability to place students in their
model. Further research is needed to examine students understanding of inverse and its impact on
31
mathematical development. In addition, research is needed to identify other concepts that may
interfere with students’ development in college mathematics courses.
Acknowledgements
Math dept and cream
32
Appendix A. Sample Student Assessment on Exponents
Name (Printed):
Date:
Assessment on Exponentials
Part A: If possible, simplify the expressions below and express your answers using positive exponents.
(4𝑥 3 )2
A.2. (−8)2/3
3
√272
A.4. −93/2
A.5.
8(−1/3)
A.6. (−4)3/2
A.7.
40 − 4−1
A.1.
A.3.
9 3/2
A.8. (4)
Part B: Compare the following expressions using the symbols <, = or >.
B.1. 238
2313
B.2.
−178
(−17)8
B.3. (−12)−8
(−12)−14
B.4.
233/5
153/5
B.5. 0.525
0.533
B.6.
115/3
3
Part C: Answer the following questions.
C.1. Why is 2𝑥+1 twice as much as 2𝑥 ?
C.2. Label each of the following numbers as either positive or negative.
(a) (−3)12
1 −4
(b) (2)
1 (−1/3)
(c) (− )
3
(d) 2−3
√115
33
Appendix B. Z-scores comparing sample proportions
Our first attempt to identify persistent errors was through 𝑍-scores, which are presented in Tables 2a, b.
These were computed to determine whether the proportions of students who correctly answered each
question was significantly different between College Algebra and Pre-calculus than those in Calculus 2.
The students in these groups were independent with a sample population size 𝑛, of at least 100 for each
group. A correction to 𝛼 = 0.05 was made because 19 tests were conducted, one for each question.
The corrected 𝛼 = 0.026 gave a critical 𝑍 value of 2.33. The test results indicated a significant
improvement between College Algebra and Calculus 2 students for all questions except B1. The test
results indicated a significant improvement between Pre-calculus and Calculus 2 students for all
questions except A1, B1, B3, and B6.
Table 2: 𝑍-scores comparing sample proportions.
Table 2.a
A1
A2
A3
A4
A5
A6
A7
A8
College Algebra and Calculus 2
3.21*
6.86*
4.48*
2.95*
7.93*
5.94*
8.67*
6.27*
Pre-calculus and Calculus 2
1.92
6.05*
3.82*
3.14*
5.22*
5.38*
5.99*
5.23*
Table 2.b
B1
B2
B3
B4
B5
B6
C2a
C2b
C2c
C2d
College Algebra and Calculus 2
0.74
4.69*
2.69*
4.22*
5.14*
3.36*
4.23*
6.07*
7.28*
6.47*
Pre-calculus and Calculus 2
0.16
3.04*
1.33
4.01*
3.02*
1.56
3.83*
3.7*
4.65*
4.07*
*𝑍-test significant for 𝛼 = 0.026.
34
References
Ayres, P. (2000). Mental effort and errors in bracket expansion tasks. In J. Bana and Al Chapman
(Eds.). Mathematics education beyond 2000. Australia: MERGA 23 (Also PME 2000).
Barcellos, A. (2005). Mathematical misconceptions of college-age algebra students. Unpublished
dissertation. University of California, Davis.
Barnes, H. (2006). Effectively using new paradigms in the teaching and learning of mathematics:
Action research in a multicultural South African classroom. Retrieved on December 15, 2010
from
http://209.85.165/search?q=cache:QFp1nAAMGuekJ:math.unipa.it/~grim/SiBarnes,PDF+Effec
tively+using+new+paradigms&hl=el&ct=clnk&cd=1
Carraher, D. & Schliemann, A. (2007). Early algebra and algebraic reasoning. In F.K. Lester (Ed.),
Second handbook of research of mathematics teaching and learning (669-706). Charlotte, NC:
Information Age.
Chalouh, L. & Herscovics, N. (1983). The problem of concatenation in early algebra. In J. Bergeron, &
N. Herscovics (Eds.), Proceedings of the fifth annual meeting of the North American Chapter of the
international group for the psychology of mathematics education: Montreal.
Chiu, C., Ibello, M., Kastner, C., & Wooldrige, J. (2009). Congitive Obstacles and common
Misconceptions. Retrieved June 28, 2010 from
http://eee.uci.edu/wiki/index.php/Quadratics#Cognitive_Obstacles_and_Common_Misconcepti
ons
Coolmath (2011). Coolmath pre-algebra. Retrieved July 13, 2011 from
http://www.coolmath.com/prealgebra/08-signed-numbers-integers/05-signed-numbers-integersopposite-01.htm.
Demby, A. (1997). Algebraic procedures used by 13-to 15-year-olds. Educational Studies in
Mathematics, 33, 45-70.
35
Dugdale, S. (1993). Functions and graphs – perspectives on student thinking. In T. P. Carpenter
(Ed.), Integrating research on the graphical representation of functions, 101-130. Hillsdale, NJ:
Erlbaum.
Indiogine, S. E. (2008). Misconceptions about Exponents. Term paper for Math 689 – Teaching
Mathematics at the College Level, Summer 2007 at Texas A&M University. Retrieved
(01/17/2011) from:
http://people.cehd.tamu.edu/~sindiogine/pdf/Misconceptions_about_Exponents.pdf
Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels. In F.K.
Lester (Ed.), Second handbook of research of mathematics teaching and learning (707-763).
Charlotte, NC: Information Age.
Knuth, E. J., Stephens, A.C, McNeil N.M., & Alibali, M. W. (2006). Does understanding of equal sign
matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4),
297-312.
Lappan, G., Fey, J.T., Fitzgerald, W.M., Friel, S.M. & Phillips, E.D. (2006). Connected Math 2: Accentuate
the negative, Integers and rational numbers. Boston: Pearson, Prentice Hall.
Lee, M.A. & Messner, S.J. (2000). Analysis of concatenations and order of operations in written
mathematics. School Science and Mathematics, 100(4), 173-180.
Liston, M. & O’Donoghue, J. (2010). Factors influencing the transition to university service
mathematics: Part 2 a qualitative study. Teaching Mathematics and Its Applications, 29, 53-68.
Matz, M. ( 1980). Towards a computational theory of algebraic competence. Journal of
Mathematical Behavior, 3(1), 93-166.
Miles, M. & Huberman, M. (1994). Qualitative data analysis: A sourcebook of new methods. Beverley
Hills: Sage.
Mullett, E. & Cheminat, Y. (1995). Estimation of exponential expressions by high school students.
Contemporary Educational Psychology, 20, 451-456.
36
Pitta-Pantazi, D. Christou, C. & Zachariades, T. (2007). Secondary school students’ level of
understanding in computing exponents. Mathematical Behavior, 26, 301-311.
Sakpakornkan, N. and Harries, T. (2003). Pupils' processes of thinking: Learning to solve algebraic
problems in England and Thailand. In J. Williams (Ed.), Proceeding of the British society for
research into learning mathematics, 23, 91-97.
Sastre, M., & Mullet, E. (1998). Evolution of the Intuitive Mastery of the Relationship between Base,
Exponent, and Number Magnitude in High-School Students. Mathematical Cognition, 4(1), 6777.
Seng, L. K. (2010). An error analysis of form 2 (grade 7) students in simplifying algebraic
expressions: A descriptive study. Education and Psychology, 8(1), 139-162.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and
objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1-36.
Sfard, A. (1992). Operational origins of mathematical objects and the quandary of reification – the
case of function. In Dubinsky E., & Harel, G. (Eds.) The concept of function: Aspects of
epistemology and pedagogy, MAA p. 54-84.
Sfard, A. & Linchevski, L. (1994). The gains and the pitfalls of reification: The case of algebra.
Educational Studies in Mathematics, 26, 191-228.
Slavit, D. (1997). An alternate route to reification of function. Educational Studies in Mathematics,
33, 259-281.Tall, D.O. & Vinner, S. (1981). Concept Image and Concept Definition in
Mathematics with Particular Reference to Limit and Continuity. Educational Studies in
Mathematics, Vol. 12, 151-169.
Thompson, P. W. (1994).
University of California Irvine (2009) Retrieved June 28, 2010 from
https://eee.uci.edu/wiki/index.php/File:Misconceptions.jpg
37
Vinner, S. (1992). The Function Concept as a Prototype for Problems in Mathematics Learning. In G.
Harel, & E. Dubinsky, (Eds) The Concept of Function: Aspects of Epistemology and Pedagogy,
MAA, 195-213.
Vlassis, J. (2002a). About the flexibility of the minus sign in solving equations. In A. Cockburn & E.
Nardi (Eds.), Proceeding of the 26th conference for the International Group of the Psychology of
Mathematics Education. Vol. 4 (321-328). Norwich, UK: University of East Anglia
Vlassis, J. (2002b). The balance model: Hindrance or support for the solving of linear equations
with one unknown. Educational Studies in Mathematics, 49(3), 341-359.
Vlassis, J. (2004). Making sense of the minus sign or becoming flexible in ‘negativity’. Learning and
Instruction, 14, 469 – 484.
Warren, E. (2003). The role of arithmetic structure in the transition from arithmetic to algebra.
Mathematics Education Research Journal, 15, 122-137.
Weber, K. (2002a). Developing Students’ Understanding of Exponents and Logarithms. In
Proceedings of the 24 annual Meeting of the North American Chapter of Mathematics Education.
http://eric.ed.gov.ERICDocs/data/ericdocs2/content_storage_01/0000000b/80/27/e8/b5.pdf
Weber, K. (2002b). Students’ understanding of exponential and logarithmic functions. Proceedings
from the 2nd International Conference on the Teaching of Mathematics.
http://www.eric.ed.gov/PDFS/ED477690.pdf