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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

A B S T R A C T

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)

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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

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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.

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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

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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.

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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, a

student might write 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,

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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 – but –

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 , where multiplication is implied. After many years of high school, some

college students misapplied multiplication to mixed numbers. They simplified

as

.

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Clearly, these students misinterpreted the meaning of

by misapplying their understanding of

. Students also misapply their understanding of when it takes the form . In this case,

they misinterpret 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, ). As a process, students can imagine the result of exponentiation

without actually performing it, . As the result of a process, exponential

expressions are viewed both as a prompt to compute and as a mathematical object that can be

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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 where

and 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 only when

and 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:

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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

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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

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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 Total No. Students

Overall Score (%)

Percent Correct by Section

Percent Correct (Section A Questions)

Section A

Section B

Section C

1 2 3 4 5 6 7 8

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

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17% of students in College Algebra correctly answered question A7, which asked students to

simplify

.

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 and ( ) . 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 was negative.

5.2. Indicators of persistent errors

Our first attempt to identify persistent errors was through -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 -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.

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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 Pitta-

Pantazi, et al (2007) summarizes student responses organized by question.

Table 3

Problems discussed during student interviews.

Section Problem Level of Conceptual

Understanding

Assessed Knowledge

A 2 ( ) 3 Simplify a negative base with a rational

exponent

A 4 3 Simplify an additive inverse with a

rational exponent

A 6 ( ) 3 Simplify a negative number with a rational

exponent

B 2

( )

1 Compare two numbers raised to a power

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C 2d 2 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 ( ) . 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 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 ( )

√

√

It appears that the student interpreted as ( ) . The other student solved it

two ways. The first approach is carried out correctly as ( ) ; √

The student’s second

approach contains the same notational error as above, that is, interpreting as ( ) . The

student wrote, (√

) .

All of the students who incorrectly simplified the expression moved the negative sign

inappropriately. One student wrote, ( ) ( ) When asked why he moved

the negative sign outside the parentheses, he responded, “You don’t need them, they are more for

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clarity.” Two other students moved the negative sign to the exponent and one wrote ( )

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

negative sign signaled that a reciprocal is involved. This student wrote ( )

and was unable to further simplify the problem.

5.3.2. Question A4

This question asked students to simplify the expression . 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 √

and remarked, “You can’t take the square root of a negative number.” Another student wrote

( ) ( ) ( ) = 729 and concluded that was equal to √ √ . 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 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 ( ) . 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

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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 ( ) and √ so . Oh,

can’t have √ , can’t do this, you get an .” 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

( ) ( )

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 ( ) ] 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 and , she said that “ and

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= 9.” In the former expression, she interpreted as subtract (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, ( ) because with parentheses multiply times an

even number of times.”

5.3.5. Question C2.d

This question asked students to label the number 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, , 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 . Another student recognized that a reciprocal was

involved. She said, “… gotta flip it over somehow, something goes under to get rid of the negative.”

She then wrote

and said, “It might be [pointing at

], but I don’t think so.” Two other students

simplified to

. One of them corrected himself and reasoned, “Well, two to the minus one is one-

half so two to the minus three is three-halves. [pause] maybe [pause] let’s see.” He wrote

;

, and then stated, “No, it’s [wrote

]. So, [wrote

].” Eight students simplified

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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 as

equal to ( ) . Here, students interpreted the negative sign as “stuck” to the nine instead of

realizing that is the additive inverse of . 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 as or or ( ) or . 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 (note the position of the negative sign) is at times used to

denote the additive inverse of the number .

6.1.1. Language

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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 as ( ) 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 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 ( ) 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 and ( ) 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 K-

12 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 is – ” (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 – could represent either a positive

or negative number or that is the opposite (additive inverse) of . 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 ). Only 29% of students in second-semester calculus correctly simplified it. Thirteen

out of eighteen students interviewed initially interpreted as ( ) . Evidently, they saw

as a signed number in which the negative sign was “stuck” to the base. That is, students view 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 , 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

( ) ( ) and ( ) ( )

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 – ( ) and ( ) are not yet understood.

When students were asked to compare to ( ) , several of the students who had

incorrectly interpreted as ( ) were able to distinguish between and ( ) .

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 , but also claimed that . In the context of , she

was able to interpret correctly but when 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 as ( ), 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 missing-

addend 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 . An

operational view of solving such equations uses the notion of inverse (opposite) operations. That is,

to undo the action of adding , the opposite operation of subtracting from both sides is employed.

This is typically written as

A structural view consists of adding the additive inverse to both sides of the equation as illustrated

below.

( ) ( )

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

( ) √

√

While he did not write √( )

, the student appeared to recognize that the quantity to be squared

was . 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

instead of

.

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 and ( ) .

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 and

( ) emerges from early instruction on signed numbers, a concept used to facilitate

understanding of negative numbers and subtraction. The notation and 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 from is usually written (notice that parentheses are not used). In

conventional mathematical notation, the expression is written, ( ). However, we encounter

college students who use nonconventional notation (e.g., ). 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., , Figure 1).

Negative numbers are usually written with a dash like a subtraction sign.

and

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.]

( )

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, ( ), leads

to the adoption of unconventional notation. The expression indicates that the

negative sign preceding the eight is “stuck” to the eight, that is, is an inseparable entity.

It also implies that the parentheses used in conventional notation are not necessary. When

26

is equated to ( ), 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 is when reminded that – is the additive inverse

of . 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 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 , some students formed the reciprocal of the exponent and wrote . Another

student formed the reciprocal of the exponent and placed it into the denominator of a rational

number and wrote

. Similarly, two other students wrote

and

. 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 . 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 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 as

repeated multiplication, . They do not see as an object that can be manipulated nor do

they see as the multiplicative inverse of . To develop a structural understanding of the

algebraic properties of numbers, students must link with ( ) .

In this study, a few students used the definition, , to remind themselves how to

simplify . Looking solely at the notation, they incorrectly generalized the definition to the

erroneous statement, . They assumed that the base forms the denominator and the

exponent forms the numerator. Applying this incorrect generalization, they wrote .

Without a structural understanding, their interiorization of the definition, , failed to

convey the notion of multiplicative inverse. We are unsure whether students’ inability to simplify

derives from their failure to see 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

reciprocal,

, and an structural understanding that is linked to inverse, 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

( ) by dividing both side by . In this operational approach, students simply divide both sides

by . A structural approach has the advantage of introducing both the terms and concept of

multiplicative inverse and multiplicative identity into the students’ discourse.

( )

( )

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 correctly, but they did not

recognize the unary operation as the additive inverse of . We suspect that student failure to

recognize 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 , where , students understood that

something needed to be “flipped,” but did not recognize it as the unary operation of the

multiplicative inverse of .

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

or ( ) . 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.

A.1. ( )

A.2. ( )

A.3. √

A.4.

A.5. ( )

A.6. ( )

A.7. A.8. (

)

Part B: Compare the following expressions using the symbols or .

B.1. B.2. ( )

B.3. ( ) ( ) B.4.

B.5. B.6. √

Part C: Answer the following questions.

C.1. Why is twice as much as ?

C.2. Label each of the following numbers as either positive or negative.

(a) ( )

(b) (

)

(c) (

)( )

(d)

33

Appendix B. Z-scores comparing sample proportions

Our first attempt to identify persistent errors was through -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 , of at least 100 for each

group. A correction to was made because 19 tests were conducted, one for each question.

The corrected gave a critical value of . 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: -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*

* -test significant for .

34

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