# Procedural and Conceptual Understanding of Rational Expressions

```Algebra Students and Rational Expressions 1
Running Head: ALGEBRA STUDENTS AND RATIONAL EXPRESSIONS
Algebra Students’ Simplification of Rational Expressions
Hanna Figueras, Lorraine Males and Samuel Otten
Michigan State University
Algebra Students and Rational Expressions 2
Algebra Students’ Simplification of Rational Expressions
Even algebra students who are adept at simplifying a variety of polynomial
expressions often face a significant challenge when confronted with rational expressions—the
quotient of two polynomials. There are many possible reasons for this. Perhaps students do
not have a solid comprehension of rational numbers and so are unable to handle something
that looks like a &quot;fraction with x's in it.&quot; Perhaps they are unsure of the order of operations
since there may be operations both &quot;above&quot; and &quot;below&quot; the operation of division. Perhaps
they have inadequate conceptual understanding of inverse operations and so inappropriately
cancel terms. Or perhaps it is something else entirely. Nevertheless, this is important because
errors in this domain, according to our experience, are frequent, and algebra students will
likely see rational expressions again later in their algebraic career (and certainly if they go on
to calculus).
The purpose of this research was to examine how algebra students simplify rational
expressions, thus contributing to the existing research on algebra learning. Although rational
expressions have not been the direct focus of much research, the domain promises to provide
a rich supply of data for several reasons. First, experience leads one to expect student errors to
be plentiful. Second, the complexity of rational expressions suggests that the strategies
students use to simplify them are likely to be diverse. Third, there are strong visual cues in
many rational expressions as objects are placed in close vertical or diagonal proximity to one
another. Fourth, rational expressions may be the first entity in algebra for which students are
not given or are unable to generate a concretization to correspond with the expression. These
Algebra Students and Rational Expressions 3
points imply that early algebra students, even those progressing successfully through the
material, may face their first serious roadblock in rational expressions.
Much of the past work regarding algebra learning has focused on student errors during
algebraic manipulation (e.g., Matz, 1980; Demby, 1997). A recurring theme in these studies
has been reliance by students on the visual cues present in algebraic structures. A visual cue is
anything in the structure of a mathematical object that brings to mind a piece of mathematical
knowledge or a memory of a mathematical experience. An illustration of this would be a
student who sees 3( x  2) and recalls the distributive property based on the visible structure
of a number situated adjacent to parentheses. Reliance on visual cues is not necessarily
undesirable since skilled algebra students can use visual cues efficiently, appropriately, and
successfully. However, visual cues may lead many algebra students to commit errors. For
example, Demby (1997) wrote about the 3 and -3 appearing in the expression 2 x  3  3x as
being visually attractive, often leading students to cancel inappropriately. Kirshner (1989)
found that some students depend on visual cues rather than propositional reasoning, and that
students who lack a propositional foundation for their syntactic knowledge often retrieve
inappropriate rules.
Rational expressions present strong visual cues, with like terms (or identical terms)
often situated vertically or diagonally from one another, as in
3x  6
.
3x
This expression may prompt students to recall the cancellation property
ab
 b . If students
a
lack a firm propositional foundation (Kirshner, 1989), then they may inappropriately cancel
Algebra Students and Rational Expressions 4
the 3x in the numerator and denominator, as was found by Matz (1980) who referred to it as
the cancellation error. It is important to note that a student committing such a cancellation
error may indeed possess the corresponding propositional knowledge but may simply have
failed to activate this knowledge in the face of such strong visual cues. Do certain
presentations of rational expressions make it more likely for a student to access propositional
knowledge during simplification? With this question in mind, we turn to literature that regards
mathematical entities as processes and objects.
According to Sfard (1991) there are two different ways in which to conceive
mathematical notions: operationally (as processes) and structurally (as objects). It is
especially difficult for early algebra students, transitioning from arithmetic to algebra, to
conceive of polynomials as objects rather than processes. For example, students may see the
expression 3( x  5)  1 as a process or a sequence of instructions, such as: add five to a given
number, multiply the result by three, and then add one to this result (Sfard &amp; Linchevski,
1994). Similarly, Gray and Tall (1992b) wrote about the proceptual divide. A procept is “a
combined mental object consisting of both process and concept in which the same
symbolization is used to denote both the process and the object which is produced by the
process” (p. 4). For example, 3  2 signifies the counting process and the product of this
process, namely, 5 . Thus, seeing 3  2 conjures up both the process of addition and the
concept of sum. According to Gray and Tall (1992), the proceptual divide occurs between
students who can “successfully see the algebraic notation as flexible procept and those who
see it only as process and fall into routines using instrumental procedures to carry out
computations” (p. 6).
Algebra Students and Rational Expressions 5
This notion of process and object can be found in many other works as well, though
often with different terminology. Freudenthal (1983) wrote about the difference between
procedural and static interpretations, the former pronouncing a  b as “a plus b” and the latter
viewing a  b as the sum of a and b. According to Kieran (1992), Matz and Davis have also
studied this idea, calling it the process-product dilemma. Regardless of the language used, this
conceptual framework has implications for rational expressions. Students who commit
cancellation errors may be viewing the rational expression as a process to be carried out, with
cancellation being an expected part of this process. Students who interpret a rational
expression (at least to some extent) as a quotient of two objects may be less likely to commit
cancellation errors because such a cancellation involves an inappropriate parsing of the
objects. In addition, there may be a relationship between the object conception of rational
expressions and the propositional knowledge needed to avoid cancellation errors. A
presentation of rational expressions that emphasizes the object interpretation, then, would be
expected to negatively correlate with cancellation errors during simplification.
Research Questions
This study examined algebra students’ simplification of rational expressions with an
emphasis on the relationship between cancellation errors and an object interpretation of
rational expressions. We focused on the following research questions:
1.
What errors do algebra students commit when simplifying rational expressions?
2.
What strategies do algebra students use to simplify rational expressions?
Algebra Students and Rational Expressions 6
3.
What is the effect on the frequency and types of errors committed when rational
expressions are presented in a way that affords an object interpretation?
Method
Study Design
To investigate students’ errors and strategies in simplifying rational expressions, we
administered a survey to groups of students in grades 9, 11 and 12. The survey comprised
two parts. Part 1 consisted of 7 items designed to determine base levels of cancellation errors
committed. Part 2 consisted of 4 items designed to present rational expressions with an
emphasis on an object interpretation. (See the appendix for a complete list of the survey
items.)
Participants
The study included 80 students. Thirty-six were 10th-grade students in a geometry
class in a rural public high school in northern Michigan. Forty-four were 11th- or 12th-grade
students in a pre-calculus class in a rural/suburban public high school in central Michigan. All
students had completed at least one year of a course designated as algebra. All had studied
rational expressions prior to the time that the survey was given, thus having had an
opportunity to learn how to simplify rational expressions.
Instruments
The survey consisted of 11 problems divided into two parts. The parts were known to
the researchers, but the orders of the problems were randomized so that neither part preceded
the other thus preventing the students from learning from previous survey items. Part 1
problems (there were 7) presented rational expressions in the variable x to be simplified. The
Algebra Students and Rational Expressions 7
coefficients and constants in the expressions were positive integers. Part 2 problems (there
were 4) presented rational expressions, each of which was designed in a different way to
evoke interpretation as mathematical objects. The first used parentheses, the second identified
the numerator and denominator with visual symbols, and the third and fourth presented
situations in which the numerator and denominator could be readily associated with
something either conceptually or visually. Each problem in Part 2 corresponded to one of the
problems in Part 1. The first two problems in Part 2 were varied so that they corresponded
with distinct problems from Part 1.
Procedure
With the permission of the principals and teachers, the survey was given cooperatively
by teachers and researchers to the students in their mathematics classrooms. The northern
surveys were administered in December of 2007 and the central surveys in January of 2008.
The students were asked to fill out the survey in pen and show their work. The use of pens
would prevent students from changing answers to earlier questions as they progressed through
the survey. The students were given approximately 20 minutes to complete the survey.
Analysis
The raw data consisted of responses to 80 surveys with 11 items in each survey. One
item was discarded from analysis (the reason for this will be discussed later in this paper). In
addition, five surveys were discarded due to a large number of uncodable items (see below for
definition of uncodable). Thus, the final analysis included a total of 750 items from 75
surveys. Categories of errors were developed based on the student work, frequencies of these
Algebra Students and Rational Expressions 8
errors were tabulated, and categorical analysis was conducted on responses to the linked items
from Part 1 and Part 2.
Categorization of Errors
Initial analysis entailed coding student responses for types of errors. We identified the
following primary error categories: cancellation (C), partial division (P), like-term error 1
(T1), like-term error 2 (T2), linearization (L), defractionalization (D) and equationization
(E). These seven categories accounted for 83% of the errors and are described in detail below,
with examples presented in Table 1. It should be noted that 24% of the items containing errors
were double-coded because participants, for example, made a visual cancellation error in the
first step of their simplification and a like-term error in the second step.
In addition to these seven main error categories we used three additional codes. The
other error (OE) category consisted of any algebraic error that was not included in the
primary categories; for example, a factorization mistake such as 5 x  15  5( x  2) or a
misconception such as
x
 x was coded as OE. An item on which the participant progressed
x
through the simplification without committing an error, whether or not it was a complete
simplification, was coded as no error (NE). Finally, blank or undecipherable response and
responses in which the participant merely rewrote the problem were categorized as uncodable
(UN).
Cancellation (C). The characteristic feature of a type C error was the visible presence
of an algebraic literate—a constant term, a variable, a coefficient, and so forth—in both the
numerator and the denominator that was cancelled by the participant. This cancellation was
usually done with a written slash, but work was also classified as C if the participant simply
Algebra Students and Rational Expressions 9
removed the visible terms in the subsequent step. Furthermore, a participant canceling terms
after rewriting the rational expression to make visible the terms to be cancelled was also
coded as C.
Partial Division (P). Type P errors were characterized by division taking place, but
only between some of the terms in the rational expression. For example, in
3x  9
 3x  3 , 9
3
was divided by 3 to yield 3, but the denominator was not divided into the 3x term. (Note that a
similar example appears in the table as C because in that instance the student rewrote the
expression to make visible the 3 with which to cancel the denominator). Another example was
4 x  20
x  20
4x
x

, in which we saw that
was reduced to , but this was not division by
12  4 x
3  4x
12
3
the entire denominator.
Like-Term Error 1 (T1). Participants committing the first like-term error performed an
operation other than division (usually subtraction) between terms in the numerator and terms
in the denominator. An instance of T1 was
20  8c
 20  6c , in which we saw the
2c
difference of the like terms 8c and 2c .
Like-Term Error 2 (T2). Participants committing the second like-term error performed
a mistaken operation with terms in the numerator or performed a mistaken operation with
terms in the denominator. This included erroneous addition of like terms or inappropriate
addition of unlike terms. For example,
7x  x
8x 2

was classified as T2.
7 x  3x
10 x 2
Algebra Students and Rational Expressions 10
Linearization (L). This error, noted in Matz (1980) as well as other places, was
characterized by the breaking up of a rational expression with a compound denominator into
two separate rational expressions, as in
4
4 4
 
.
8  4x
8 4x
Defractionalization (D). This error is characterized by the transformation of a fraction
with unitary numerator to a nonfraction, such as
1
1
 x or  3 .
x
3
Equationization (E). Transforming a rational expression into a rational equation was
classified as E. For example, a participant operating within this category would convert the
expression to an equation and then conclude their work with x  3 . Note that merely writing
the equals sign between steps in the simplification process was not classified as E. The
inclusion of the equals sign had to alter the content of the item to warrant an E classification,
such as setting the rational expression equal to 0.
Not having anticipated an error where students subtract individual terms in the
denominator from individual terms in the numerator, we had created the item
expression was often incorrectly simplified as follows:
10  4x
. This
2x
10  4x
 10  2x . According to our
2x

coding scheme, this error could be categorized as T1 or P because, based on our analysis of

4
other survey items, students could be either subtracting ( 4  2  2 ) or dividing (  2 ).
2
Therefore, this item was discarded from analysis.

Results

We found a larger quantity and a wider variety of errors in the survey responses than
we had expected. In fact, 79% of all 750 items were not simplified correctly and 74% of the
Algebra Students and Rational Expressions 11
attempted items contained at least one error. Table 2 displays the frequencies of error types
across all items. As we had anticipated, the most common error was the cancellation error
(24% of all codes). In fact, 51 of the 75 participants committed the cancellation error. Partial
division and the unexpected like-term error 1 were also relatively common. Forty participants
made the partial division error, and interestingly, 38 of these 40 students also committed the
cancellation error. Also, many students made the same error on several items: 21 students
(28%) committed errors of the same type on at least 7 of the 10 items, and 33 (44%)
committed errors of the same type on at least 6 items.
Because of Part 2’s focus on object interpretation of expressions, we had anticipated
fewer errors in Part 2 than in Part 1. Part 2, however, contained a large proportion of
uncodable items (36%), whereas Part 1 only contained 8% uncodable (see Table 3). This
made it difficult to draw any conclusions on how the construction of Part 2 problems might
have afforded an object interpretation and how this correlated with simplification success.
Table 4 presents a chi-square frequency distribution of the responses to the linked items.
Based on McNemar’s test for marginal homogeneity, the only significant conclusion that can
be drawn from the data is that students who committed an error in Part 1 were likely to either
make an error or not attempt the corresponding item in Part 2 (p&lt;0.00001). A large portion of
the students simply left the word problems blank.
Discussion
We examined high-school algebra students’ simplification of rational expressions and
categorized their errors. Students were less successful in their simplifications than we
Algebra Students and Rational Expressions 12
anticipated, even though all had received prior instruction on simplification of rational
expressions. Nearly four-fifths of the items were not simplified correctly.
There was evidence that students were using consistent strategies as they worked to
simplify the rational expressions, but these strategies were most often incorrect. As was
predicted, we did see an apparent reliance on visual cues, for instance, when students would
strike out identical terms during a cancellation error or would rewrite a term so that they could
see a factor and cancel it across the division bar. Additionally, students who consistently
committed the cancellation error, partial division, or the linearization error may have been
using a strategy that was based on a flawed conception of division. The existence of a
commonality behind some of these error types was evidenced by the fact that 95% of the
participants who committed partial division also committed the cancellation error. However,
our study design did not allow us to draw conclusions regarding the thought processes
underlying the apparent strategies.
One thing that we did not anticipate but that showed up very powerfully in the data
were strategies based on a fixation with like terms. This was seen most clearly in the large
number of like-term 1 errors wherein students performed operations only between like terms,
whether or not the terms were together in the numerator or denominator, and often students’
work was completely void of any division operation. In other cases (i.e., partial division),
students performed division but only between like terms rather than between the entire
numerator and denominator. This suggests that partial division may be at the intersection of a
misconception of division and an over-attention to like terms.
Algebra Students and Rational Expressions 13
Our attempt to draw a comparison between the Part 1 and Part 2 items failed due to an
apparent lack of engagement in Part 2. This may have been a deficiency of the paper-andpencil format we implemented. A design that included a clinical interview may have been
more likely to elicit responses on problems like those in Part 2. Moreover, such a design
would allow for information to be gathered regarding the thought processes underlying the
errors and strategies that we observed. It would also be beneficial for future studies to include
the like term errors in their theoretical framing, since our data pointed significantly to this
phenomenon.
For teachers, the primary implication of this study is that high school algebra students
have difficulty simplifying rational expressions. This is especially significant because
simplification is a first step in working with rational expressions, and errors in this domain
will make it nearly impossible for students to be successful in subsequent work, for instance,
solving rational equations or rational inequalities. Teachers need to be aware of the common
errors that students make when simplifying rational expressions, especially the cancellation
error since this may be frequent. This study also suggests that teachers be aware of their
emphasis on like terms during instruction. Though it is clearly necessary to cultivate in
algebra students an awareness of like terms, it is also equally necessary for them to
understand when and when not to use like terms as the basis for simplification strategies.
Algebra Students and Rational Expressions 14
References
Demby, A. (1997). Algebraic procedures used by 13-to-15-year-olds. Educational Studies in
Mathematics, 33(1), 45-70.
Freudenthal, H. (1983). Didactical Phenomenology of Mathematical Structures. Dordrecht:
D. Reidel Publishing Company.
Gray &amp; Tall (1992). Success and failure in mathematics: Procept and procedure. In Workshop
on Mathematics Education and Computers, (pp. 209-21), Taipei: Taipei National
Kieran, C. (1992). The learning and teaching of school algebra. In D. Grouws (Ed.),
Handbook of research on mathematics teaching and learning, (pp. 390-419). New York:
MacMillan Publishing Company.
Kirshner, D. (1989). The visual syntax of algebra. Journal for Research in Mathematics
Education, 20(3), 174-187.
Linchevski, L. &amp; Livneh, D. (1999). Structure Sense: The relationship between algebraic and
numerical contexts. Educational Studies in Mathematics, 40(2), 173-196.
Matz, M. (1980). Towards a computational theory of algebraic competence. Journal of
Mathematical Behavior, 3, 93-166.
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), 1-36.
Algebra Students and Rational Expressions 15
Sfard, A. &amp; Linchevski, L. (1994). The Gains and the Pitfalls of Reification: The Case of
Algebra. Educational Studies in Mathematics, 26 (2/3), 191-228.
Algebra Students and Rational Expressions 16
Table 1
Examples of student errors
Error
Student Examples
Categories
Cancellation
Partial
Division
Like-Term 1
Like-Term 2
Linearization
&amp; Other Error
Algebra Students and Rational Expressions 17
Table 2
Error Frequencies
Error
Frequency (n = 557)
Cancellation error (C)
0.24
Partial division (P)
0.16
Like-term 1 (T1)
0.19
Like-term 2 (T2)
0.08
Linearization (L)
0.09
Defractionalization (D)
0.05
Equationization (E)
0.03
Other error (OE)
0.17
Algebra Students and Rational Expressions 18
Table 3
Comparison of the Two Parts
Part I
Part II
n = 450 items
n = 300 items
371 error codes
186 error codes
No error (NE)
0.24
0.17
Uncodable (UN)
0.08
0.36
0.76
0.83
0.74
0.74
Erroneous or uncodable items
(over total)
Erroneous items (over attempted)
Algebra Students and Rational Expressions 19
Table 4
Chi-square distribution of the 225 linked items
Part 1
No error
Errors
Uncodable
No error
32
8
2
Errors
9
96
5
Uncodable
7
56
10
Part 2
```