Journal of Mathematical Behavior 28 (2009) 68–77
Contents lists available at ScienceDirect
The Journal of Mathematical Behavior
journal homepage: www.elsevier.com/locate/jmathb
Undergraduate mathematics majors’ writing performance producing
proofs and counterexamples about continuous functions
Yi-Yin Ko ∗ , Eric Knuth
University of Wisconsin-Madison, 225N. Mills Street, Madison, WI 53706, United States
a r t i c l e
i n f o
Keywords:
Counterexample
Proof
Continuous functions
a b s t r a c t
In advanced mathematical thinking, proving and refuting are crucial abilities to demonstrate whether and why a proposition is true or false. Learning proofs and counterexamples
within the domain of continuous functions is important because students encounter continuous functions in many mathematics courses. Recently, a growing number of studies have
provided evidence that students have difficulty with mathematical proofs. Few of these
research studies, however, have focused on undergraduates’ abilities to produce proofs
and counterexamples in the domain of continuous functions. The goal of this study is to
contribute to research on student productions of proofs and counterexamples and to identify their abilities and mathematical understandings. The findings suggest more attention
should be paid to teaching and learning proofs and counterexamples, as participants showed
difficulty in writing these statements. More importantly, the analysis provides insight into
the design of curriculum and instruction that may improve undergraduates’ learning in
advanced mathematics courses.
Published by Elsevier Inc.
1. Introduction
Proving and refuting are crucial abilities in advanced mathematical thinking because they help demonstrate whether
and why propositions are true or false. In the mathematics community, proving and refuting are inextricably linked given
the role each plays in establishing mathematical knowledge (Lakatos, 1976). A mathematical proof requires that definitions,
statements, or procedures are used to “deduce the truth of one statement from another” (Tall, 1989, p. 30), helping people
understand the logic behind a statement and “insight into how and why it works” (Tall, 1992, p. 506). Counterexamples similarly play a significant role in mathematics by illustrating why a mathematical proposition is false; a single counterexample
is sufficient to refute the falsity of statements (Peled & Zaslavsky, 1997). Taken together, mathematical proofs and counterexamples can provide students with insight into meanings behind statements and also help them see why statements are true
or false. Accordingly, undergraduate students in advanced mathematics are expected to learn and to use both proofs and
counterexamples throughout the undergraduate mathematics curriculum.
Before constructing a proof for a true statement or generating a counterexample for a false one, students and teachers need
to be able to accurately decide the truth or falsity of a given proposition. Research investigating undergraduate students’ and
mathematics teachers’ ability to evaluate a given proposition, however, suggest that many of them have difficulty verifying the
truth and falsehood of given statements due to their inadequate understanding of the mathematical content (Barkai, Tsamir,
Tirosh, & Dreyfus, 2002; Riley, 2003). Despite the importance of teaching and learning proofs and counterexamples, Thurston
∗ Corresponding author. Tel.: +1 608 265 5106; fax: +1 608 263 1039.
E-mail address: [email protected] (Y.-Y. Ko).
0732-3123/$ – see front matter. Published by Elsevier Inc.
doi:10.1016/j.jmathb.2009.04.005
Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
69
(1994) found that mathematicians often struggle with explaining how to write formally complete mathematical proofs to
students and concluded that more effective ways of teaching these are needed. Current research also supports Thurston’s
conclusions: studies have shown that many undergraduates and mathematics teachers who have completed several advanced
mathematics courses at the collegiate level or who have received a bachelor’s or master’s degree in mathematics still have
difficulty with proofs (e.g., Cusi & Malara, 2007; Goetting, 1995; Harel & Sowder, 1998; Knuth, 1999, 2002a, 2002b; Martin
& Harel, 1989; Moore, 1990, 1994; Morris, 2002; Stylianides, Stylianides, & Philippou, 2004, 2007; Weber, 2001, 2004) and
counterexamples (e.g., Barkai et al., 2002; Peled & Zaslavsky, 1997; Zaslavsky & Peled, 1996). When facing mathematical
proofs, many undergraduate students seem to lack adequate understandings of the components of mathematical proofs
(e.g., Harel & Sowder, 1998; Martin & Harel, 1989), or have insufficient conceptual understandings of writing mathematical
proofs (e.g., Moore, 1990, 1994; Weber, 2001, 2004).
Research also has documented undergraduates’ difficulties coordinating their informal and formal understandings of the
concept about continuous functions (e.g., Bezuidenhout, 2001; Ferrini-Mundy & Graham, 1994; Lauten, Graham, & FerriniMundy, 1994; Shipley, 1999; Tall & Vinner, 1981; Vinner, 1992; Wilson, 1994; Williams, 1991), which is essential content
across the world in college mathematics (as well as in pre-calculus and calculus courses in high school). Findings indicate that
college students have difficulty connecting the ideas of continuity and functions (Lauten et al., 1994; Vinner, 1992; Wilson,
1994), use their partially correct concept image – defined as mental pictures – to think about continuity1 (Ferrini-Mundy &
Graham, 1994; Tall & Vinner, 1981), and possess inadequate understandings between continuity and limits (Bezuidenhout,
2001; Williams, 1991). Even though undergraduates currently learn an important theorem in the domain of continuous
Functions – the Intermediate Value Theorem – in a class with a focus on writing mathematical proofs, some of them are
still unable to provide a valid proof for the Intermediate Value Theorem because they do not understand the proof for that
theorem (Shipley, 1999).
Yet despite the importance of proofs and counterexamples in undergraduate mathematics and the difficulty students
have producing and comprehending proofs and counterexamples, few studies have focused specifically on students’ abilities
to produce proofs and counterexamples in the domain of continuous functions—a domain that is both central to and pervasive in undergraduate mathematics and that students have learned in their previous calculus course and currently learn
in their advanced calculus. The main purpose of this study is to examine undergraduate mathematics majors’ performance
constructing proofs and generating counterexamples This study was guided by two research questions: (1) How well do
undergraduates construct proofs and generate counterexamples in the domain of continuous functions? (2) What problems
appear in the proofs students construct or the counterexamples they generate? We hypothesized that the majority of participants were able to evaluate given propositions correctly as well as to produce complete proofs and counterexamples
about continuous functions since they all had studied the topic. The results reported in this article focused on undergraduate
mathematics majors’ written responses regarding proofs and counterexamples about continuous functions.
2. Conceptions of proof
From a traditional perspective, “a mathematical proof is a formal and logical line of reasoning that begins with a set of
axioms and moves through logical steps to a conclusion” (Griffiths, 2000, p. 2). Stylianides (2007) defined proof to include
the essential components of sets of accepted statements, modes of argumentation, and modes of argument representation. In
this definition, proof serves as a means to communicate thoughts with learners in the mathematics community. Common
sense suggests that individuals who understand what constitutes a mathematical proof may be more successful at evaluating
purported arguments or their written responses as a valid proof or not. Indeed, “[a] person’s (or a community’s) proof scheme
consists of what constitutes ascertaining and persuading for that person (or community)” (Harel & Sowder, 2007, p. 809);
therefore, Harel and Sowder (1998) provided proof schemes in order to identify students’ individual proof work which they
are convinced.
In order to better characterize undergraduates’ proof productions for a true proposition, three proof classifications from
Harel and Sowder’s (1998) framework applied to the study reported here. The first proof category, the inductive proof scheme,
describes how individuals convince themselves or persuade others by providing one or more particular examples, which
corresponds to Balacheff’s (1988) naive empiricism (verification by several randomly selected cases) and crucial experiment
(verification by carefully selected cases). Finlow-Bates, Lerman, and Morgan (1993) and Healy and Hoyles (2000) used a
similar term, empirical, to indicate that students produce examples as proofs to convince themselves.
The second proof category, non-referential symbolic proof scheme, demonstrates that individuals employ symbolic
manipulations with little or no coherent understandings of their meanings. In other words, students manipulate symbols
with no “functional or quantitative reference[s]” (Harel & Sowder, 1998, p. 250). The third proof category, the structural proof
scheme, suggests that individuals realize that “definitions and theorems belong in the structure created by a particular set
of axioms” (Knapp, 2006, p. 28). This proof scheme is similar to Balacheff’s (1988) calculations on statements and Weber’s
(2004) and Weber and Alcock’s (2004) syntactic proof productions. According to Balacheff (1998), calculations on statements
mean students rely on definitions, theorems, or explicit properties related to the statement when producing a proof. Similarly, Weber (2004) and Weber and Alcock (2004) described how an individual attempts to construct a proof by stating the
1
Continuity in this paper refers to the continuity of functions.
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Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
mathematical definition or using related facts that he or she knows about the concepts of producing proofs as defined by
syntactic proof productions.
3. Conceptions of counterexample
Although the ways students generate counterexamples have traditionally been connected to false conjectures (Bills et al.,
2006) or examples in the process of proving (Zaslavsky & Ron, 1998), some researchers have provided different types of categories to identify students’ counterexamples. Such categories help demonstrate if students’ counterexample productions are
sufficient to refute the falsehood of statements and what possible difficulties students may have. We use the following categories, based on Peled and Zaslavsky’s (1997) and Zaslavsky and Ron’s research (1998), to describe undergraduate students’
counterexample writing abilities.
Peled and Zaslavsky (1997) identified an inadequate counterexample, which failed to refute the false claim, and an adequate counterexample, which succeeded in refuting the false claim. Zaslavsky and Ron (1998) used similar terms: an incorrect
counterexample, which was insufficient for rejecting the false proposition; a correct counterexample, which was successful
in rejecting the false proposition; and justification (without giving a counterexample) to classify students’ written responses
regarding counterexamples. Peled and Zaslavsky (1997) and Zaslavsky and Ron (1998) found that some participants provided
either an example which did not satisfy the condition for a counterexample or a counterexample which did not exist. In order
to better characterize what undergraduates viewed their counterexamples as sufficient productions for a false proposition,
the results reported in this article are based on the above counterexample taxonomy.
4. Methods
4.1. Participants
Thirty-six Taiwanese undergraduates enrolled in Advanced Calculus I in Fall 2007 at a national university in Taiwan
participated in this study. They were selected by convenience sampling; that is, participants were contacted by colleagues
of the researchers and were recruited on the basis of their willingness to participate in the study. Every undergraduate who
volunteered for the study was accepted.
4.2. Participants’ backgrounds
Some participants were taught by one instructor who had 28 years of experience teaching mathematics at the university
where this study took place, and the remaining participants were taught by an instructor with 9 years of experience. All
participants were taught by two different teaching assistants, who were graduate students in mathematics, during their
discussion sections. The textbook for the course was Wade’s An Introduction to Analysis (Wade, 2003), and topics from both
Advanced Calculus I sections included the real number system, sequences and continuity in R, and differentiability and
integrability on R. Both instructors received their doctoral degrees in mathematics and used traditional teaching styles: the
instructor lectured and the students passively took notes. The course met in two 100-min and one 50-min lectures and
one 100-min discussion per week over the course of an 18-week semester. Homework was assigned by the instructor and
collected and discussed by the teaching assistant each week.
The prerequisite for taking Advanced Calculus I was passing grades in Calculus I and II. Since continuous functions were
addressed in a previous calculus course, all of the students participating in this study had some relevant domain knowledge.
Students in Advanced Calculus I were mostly between 20 and 22 years old and disproportionately male. Since this study
does not target participants based on age, gender, or other characteristics, such characteristics were not considered in the
analysis of the data.
4.3. Instrument
The instrument was written in English because English is used in advanced calculus courses at this university. The instrument, comprised of five mathematical statements that were modified from textbooks and entrance examinations, was
designed to assess students’ abilities to produce proofs and counterexamples about continuous functions (see Table 1). The
mathematical statements were designed to (a) require an understanding of continuous functions, (b) represent basic types of
proofs and counterexamples about continuous functions, and (c) be completed by each participant in approximately 30 min.
The instrument was finalized after pilot testing with Taiwanese undergraduate and graduate mathematics majors.
4.4. Data collection
The primary source of data was students’ written responses to the instrument as listed in Table 1. The instrument was
administered to the students in their advanced calculus classes after completing all course instruction about continuous
functions. Students were asked to construct proofs for statements they believed to be true and to generate counterex-
Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
71
Table 1
Five propositions used in this study
Problem
Mathematical statement
True or false
1
Let f and g be functions defined on a set of numbers S, and let a ∈ S. If f is
continuous at a and g is discontinuous at a, then fg is discontinuous at a
Let f be a function defined on a set of numbers S, and let |f| be the function whose
value at x is |f(x)|. If f is continuous at a ∈ S, then |f(x)| is continuous at a ∈ S
Let f2 be a function defined on a set of numbers S, and let a ∈ S. If f2 is continuous at
a, then f is continuous at a
Let f be a continuous function from [0,1] onto [0,1]. Then there exists a point
x0 ∈ [0,1] such that f(x0 ) = x0
False
2
3
4
Let D = [0,1] ∪ (2, 3] and define f: D → R by f (x) =
5
x
if 0 ≤ x ≤ 1
x−1
if 2 < x ≤ 3
, then
True
False
True
True
f : D → R is continuous (adopted from Fitzpatrick, 1996, p. 56)
amples for statements they believed to be false. Students were asked not to include their names in order to maintain
anonymity.
4.5. Data analysis
Data were gathered on proofs and counterexamples produced by participants. In order to better characterize undergraduates’ performance constructing proofs, we used the aforementioned proof categories as well as the categories No Response,
Restatement, Counterexample, and Completeness, as listed in Table 2, to assess the participants’ written responses to the
true statements. In order to better characterize undergraduates’ performance generating counterexamples, we used the
aforementioned counterexample categories as well as the categories No Response and Proof, as listed in Table 3, to assess
the participants’ written responses to the false statements.
The manifested errors in the students’ attempts to construct proofs and to generate counterexamples were investigated
by analyzing students’ written work. To check the reliability of the coding, two coders worked independently using eight
randomly selected responses. The coded samples were then compared and disagreements were discussed until the problems
were resolved. Data were then re-coded, taking into account any changes made to the coding scheme.
Table 2
Seven types of proof productions
Production
Description
No response
Restatement
Counterexample
Empirical
Non-referential symbolic
Left blank, no relevant knowledge, presented as a guess
Restated the problem with students’ own language but no basis for constructing a proof
Gave an incorrect counterexample attempts to refute a true proposition
Used examples as demonstrations
Manipulated symbols behind the meanings involved in problem situations with logical errors
but did not produce a proof
Presented mathematical definitions, relevant axioms or theorems that could construct a valid
proof but making logical errors
Provided a complete proof
Structural
Completeness
Table 3
Six types of counterexample productions
Production
Description
No response
Proof
Inadequate
Justification
Incomplete
Adequate
Left blank or no relevant knowledge presented as a guess
Gave an incorrect proof attempts to prove a false proposition
Provided a counterexample that failed to refute a false proposition or did not exist
Narrated a proposition that was false instead of providing a counterexample to refute it
Provided a counterexample that succeeded by refuting a false proposition but making logical errors
Provided a complete counterexample
5. Results
Three types of results are reported in this section: (1) quantitative data of students’ written responses, (2) errors manifested in students’ written work, and (3) students’ understandings of underlying concepts of continuity. We first focus on
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Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
Table 4
Students’ proof productions
Problem number
2
4
5
Complete
No response
Restatement
Counter-example Empirical
Symbolic
n
%
n
%
n
%
n
%
n
%
n
%
n
Structural
%
0
0
0
0
0
0
18
13
7
50
36
19
3
11
20
8
31
56
4
4
1
11
11
3
1
1
0
3
3
0
9
7
8
25
19
22
1
0
0
3
0
0
the quantitative data of undergraduates’ responses, and then we examine the errors and understandings that appeared in
students’ written work.
5.1. Quantitative data of undergraduates’ written responses
Illustrating students’ performance in constructing a proof or in attempting to find a counterexample for true statements
in continuous functions (assessment items 2, 4, and 5), Table 4 shows none of the students provided a complete proof for
the true mathematical statements.
The following examples illustrate proof or counterexample productions that consisted of (a) a restatement – only restating
the problem yields an invalid argument for proving, (b) a counterexample – giving an incorrect proof attempts to prove a false
proposition, and (c) non-referential symbolic—manipulating symbols behind their meanings involving problem situations
but making logical errors (Problem 2).
Problem 2. Let f be a function defined on a set of numbers S, and let |f| be the function whose value at x is |f(x)|. If f is
continuous at a ∈ S, then |f(x)| is continuous at a ∈ S.
(a) Restatement
This example is representative of three students who only restated Problem 2. This particular student indicated that
Problem 2 is a true statement, but he or she only restated the problem and did not display any relevant knowledge of
continuous functions for proving. In other words, this student seemed to believe that his or her restatement of a given
proposition counted as a proof. Therefore, this student’s proof response was coded as a restatement.
(b) Counterexample
Four students provided an incorrect counterexample for Problem 2, a true proposition; the example above shows the
work of a student who provided an incorrect counterexample. In fact, f(x) = sin x and |f(x)| = |sin x| are both continuous at
0. However, this student indicated that |f(x)| = |sin x| is not continuous at 0. That is, this student showed an inadequate
understanding of continuous functions and refuted a true proposition by using an incorrect counterexample. Thus, this
student’s work was coded as a counterexample.
Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
73
Table 5
Students’ performance generating counterexamples
Problem
1
3
No response
Proof
Inadequate
Justification
Incomplete
Adequate
n
%
n
%
n
%
n
%
n
%
n
%
10
16
28
44
14
6
39
17
0
2
0
6
0
3
0
8
3
2
8
6
9
7
25
19
(c) Non-referential symbolic
The above example is representative of the work of nine students who all attempted to manipulate symbols of limits and
absolute values of continuous functions when constructing a mathematical proof. As to the definitions of continuous functions, we cannot get lim |f (x)| = |f (a+ )| = f (a) = |f (a− )| = lim |f (x)| if f(x) is continuous at a. This student not only exhibited
x→a+
x→a−
an incorrect understanding of the symbols of limits and absolute values, resulting in an invalid proof, but also seemed to
view his or her response as a mathematical proof. So his or her work was coded as non-referential symbolic.
As evidence of students’ performance in generating a counterexample or in attempting to produce a proof for false statements about continuous functions, Table 5 indicates that only nine and seven students generated complete counterexamples
for Problems 1 and 3, respectively.
The following examples are for (a) a proof—giving an incorrect proof attempts to prove a false proposition, (b)
incomplete—a counterexample succeeding in refuting a false proposition but making logical errors, and (c) adequate—a
completely correct counterexample, when students generated a counterexample for Problem 3.
Problem 3. Let f2 be a function defined on a set of numbers S, and let a ∈ S. If f2 is continuous at a, then f is continuous at a.
(a) Proof
The example given above is representative of the responses from six students who provided an incorrect proof with
inadequate understandings of continuous functions to prove a false proposition. According to the formal definition of
continuous functions, we cannot get limf 2 (x) = f (a) if f2 is continuous at a. Also, this student did not seem to understand
x→a
the meaning of f2 is continuous at a. Hence, this student’s response was coded as a proof.
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(b) Incomplete
No,
Counter example :
1
1
, and f (x) = √
Let f 2 (x) =
(x − 1)
x−1
suppose a = (−1)
1
1
1
f 2 (a) =
=
,
lim f 2 (x) =
−2
−2
(−1) − 1
a→−1
1
1
(not exists in xy plane)
but f (a) = √
=
2i
−2
The example given above is representative of the responses from two students who almost provided a complete
2
2
counterexample to refute the false statement successfully.
In
fact, f : (−∞, 1) ∪ (1, ∞) → defined by f (x) = 1/(x − 1)
√
x − 1 is continuous. This student seemed to provide a correct
is continuous, and f : (1, ∞) → defined by f (x) = 1/
function to refute the false proposition; however, he or she forgot to provide an adequate domain for the function as well
as with an explanation of this function which was continuous. Therefore, this student’s response was coded as incomplete
(c) Adequate
The above example is representative of the work of seven students who all provided a complete counterexample to
√
2
refute a false
√claim successfully. In fact, f (x) = x is continuous at −5 belonging to R, but f (x) = x is not continuous at
−5 because −5 does not belong to R. Therefore, this student provided clear explanations for why this counterexample
succeeded in refuting the Problem 3. So this student’s work was coded as adequate.
5.2. Errors manifested in students’ written work
Errors manifested in students’ written work were analyzed in detail for each proof and counterexample. This section
reviews in greater depth a few interesting cases that illustrate students’ understanding (or lack thereof). We have chosen to
analyze the following errors because they were the ones most commonly committed by students while producing proofs
and counterexamples.
5.2.1. Problem 1
Problem 1 is a false statement, but 14 students believed this was true and attempted to construct a proof. All of
them demonstrated that “f is continuous at a ⇒ limf (x) = f (a) and g is discontinuous at a ⇒ limg(x) =
/ g(a). Because
x→a
x→a
/ lim f (x) lim g(x) = lim f (x) g(x), fg is discontinuous at a.” These students showed their
lim f (x) g(x) = lim f (x) lim g(x) =
x→a+
x→a+
x→a+
x→a−
x→a−
x→a−
partial understanding of limits related to continuous functions (i.e., If f is continuous at a, then limf (x) = f (a)) and disconx→a
tinuous functions (i.e., If g is discontinuous at a, then limg(x) =
/ g(a)). Indeed, the three conditions of a function f which is
x→a
continuous at a ∈ R are the following: (1) f(a) exists, (2) lim f (x) = lim f (x), and (3) limf (x) = f (a). Even though g is disx→a+
x→a−
x→a
/ lim g(x). Moreover, the product rule of limit functions (i.e.,
continuous at a ∈ R, it does not necessary mean that lim g(x) =
x→a+
x→a−
lim(f · g)(x) = l · m) holds if limf (x) = l and limg(x) = m where l and m belong to R. As seen above, possessing an unsatisfactory
x→a
x→a
x→a
understanding of limits and continuity led students to result in the false proposition to be true.
5.2.2. Problem 2
Problem 2 is a true proposition, for which nine students provided symbolic manipulations related to continuous
functions
but not a mathematical proof. One common problem is that “if f is continuous at a, then limf (x) = f (a) ⇒ lim f (x) = |f (a)| ⇒
x→a
x→a
Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
75
lim|f (x)| = |f (a)| ⇒ |f (x)| is continuous at a,” which three students demonstrated in their written work. These students merely
x→a
demonstrated symbolic manipulations of absolute values rather than their meanings for constructing a proof, and seemed
to believe such proof production is a valid mathematical proof.
The other is that “y = f(x) ⇒ |y| = |f(x)|. If y is continuous at a ∈ S, then lim |f (x)| = |f (a+ )| = f (a) and lim |f (x)| = |f (a− )| =
x→a+
x→a−
f (a) ⇒ |f (a+ )| = |f (a− )| ⇒ |f (x)| is continuous at a ∈ S,” in which five students attempted to manipulate symbols when
producing a proof. Even though f(x) is continuous at a ∈ S, lim |f (x)| = |f (a+ )| and lim |f (x)| = |f (a− )|, we cannot get
x→a+
x→a−
lim |f (x)| = |f (a+ )| = f (a) = |f (a− )| = lim |f (x)| according to the definition of continuity and the absolute value.
x→a+
x→a−
The remaining student demonstrated that f is continuous at a ∈ S means “for every ε > 0, ∃ı > 00 < d(x, a) < ı, lim(f(x),
f(a)) < ε” in order to produce a proof. Indeed, the formal definition of a function f is continuous at a ∈ S is for every positive
ε, there exists a positive ı, such that, for all x ∈ S, if |x − a| < ı, then |f(x) − f(a)| < ε. Thus, this student only possessed a partial
understanding with his or her own language of the definition of continuous functions.
The above evidence suggests that such students displayed a partially correct understanding of symbols of limits and absolute values related to continuous functions but did not construct a mathematical proof involved in the problem’s situations.
Therefore, their written work was coded as non- referential symbolic.
5.2.3. Problem 3
Problem 3 is a false statement, but six students believed it was true and attempted to construct a mathematical proof.
2
2
2
One general error that appeared on
such students’
written work is “f is continuous at a, then ∀ε > 0, ∃ı > 0|f (x) − f (a)| <
ε whenever |x − a| < ı. Therefore, f (x) − f (a) f (x) + f (a) < ε ⇒ f is continuous at a.” The other is “f2 is continuous at a,
so f2 (x) → f2 (a) as x → a ⇒ f(x) → f(a) as x → a.” Regarding “|f(x) − f(a)||f(x)+f(a)| < ␧” or “f2 (x) → f2 (a)” as x → a, there was no
supportive evidence to show that why f is continuous at a. The above descriptions illustrate that these students seemed to
believe that f2 is continuous at a which directly implies f is also continuous at a. They were manipulating symbols without
thinking completely so that they incorrectly verified this proposition to be true.
5.2.4. Problem 4
Problem 4 is a true proposition, and 11 students restated this problem with their own language and did not provide any
basis for constructing a mathematical proof. All of them indicated that “f is continuous from [0,1] onto [0,1], x0 ∈ [0,1], then by
the definition of continuous functions, lim f (x) = x0 = lim f (x). Therefore, there exists a point x0 ∈ [0,1] such that f(x0 ) = y0 .”
x→x+
0
x→x−
0
In fact, an onto function f from [0,1] to [0,1] is defined as for every y in [0,1], there is an x in [0,1] such that f(x) = y. Also, a
function f is continuous at x0 ∈ [0,1] which does not necessarily imply that f(x0 ) = l where l belongs to R. The above evidence
shows that these students assumed the result correctly and restated the proposition without displaying relevant knowledge
about continuous functions for producing a mathematical proof, which resulted in their written responses as a restatement.
5.2.5. Problem 5
As in Problem 4, 20 students restated Problem 5 by drawing a number line with simple descriptions, but did not provide
relevant knowledge for constructing a mathematical proof. The common problem exhibited by such students is providing
a number line with few descriptions: “ lim f (x) = 0 = lim f (x), lim f (x) = 1 = lim f (x), lim f (x) = 2 = lim f (x), then f is a
0+ →0
0− →0
1+ →1
1− →1
3+ →3
3− →3
continuous function,” which resulted in no basis for constructing a mathematical proof. From these responses, it is unclear
that why students believed their descriptions could prove the truth of this proposition. Because such students only restated
the problem instead of providing relevant knowledge about continuous functions for constructing a mathematical proof,
their written work was coded as a restatement.
6. Discussion
The participants in this study had considerable difficulty producing proofs and counterexamples, even though they completed all course instruction about continuous functions in Advanced Calculus I and had studied the topic in a previous
calculus course. The proof and counterexample productions in this study reveal that undergraduate mathematics majors
had insufficient understandings of continuous functions for determining the validity of a given statement and producing
proofs and counterexamples. Such evidence may encourage mathematics instructors to foster students’ mathematical reasoning and understanding by providing a learning environment in which students can engage in the process of verifying the
truth and falsity of statements and producing proofs and counterexamples.
Our overall findings indicate that producing a complete mathematical proof seemed to be difficult for the majority of
participants, which is consistent with Tall’s (1998) suggestion that formal proof is only appropriate for some students. Tall
(1998) further suggests that mathematics instructors should consider using various forms of proofs and counterexamples
that are meaningful to undergraduates, because ignoring students’ intuitive representations of proof does not promote their
mathematical understanding (Fishbein, 1982). Researchers, however, have argued that some mathematics instructors tend
to focus more on the format of proof and counterexample than students’ mathematical thinking (Weber & Alcock, 2009),
which may lead students to have difficulty with proof and counterexample continuously. In order to refine undergraduates’
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Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
mathematical reasoning and abilities to represent valid proofs and counterexamples, mathematics instructors may draw
attention to students’ individual logical flaws and errors explicitly, instead of directing them to formal proofs and correct
counterexamples (Weber & Alcock, 2009). Such changes may help undergraduate mathematics majors, who could become
future secondary school mathematics teachers, learn proofs and counterexamples with strong enough understandings to
implement current reform recommendations regarding proof and counterexample suggested by National Council of Teachers
of Mathematics (NCTM, 2000). We also found that generating a correct counterexample seemed slightly easier for students
than constructing a correct proof. The percentages of students able to provide correct counterexamples for Problem 1 (25%)
and Problem 3 (19%) were much higher than the percentages of students able to produce complete proofs for Problem 2 (0%),
Problem 4 (0%), and Problem 5 (0%), although the percentages were not very large in either case.
In order to determine if statements are true or false, students should have a deep understanding of concepts in the specific domain because mathematical propositions involve reasoning about mathematical content. The results reported in this
article show that participants seemed to posses unsatisfactory understandings of limits, onto functions, and discontinuous
and continuous functions, which led them to verify given statements incorrectly. To enhance undergraduate students’ understandings of mathematical concepts, mathematics instructors should be aware of students’ misconceptions (Bezuidenhout,
2001) and devote more time to interpreting concepts (Vinner, 1992). Since providing opportunities for students to engage
in verifying the truth and falsity of propositions is one way to promote their understanding of and reasoning about mathematical knowledge (Buchbinder & Zaslavsky, 2007), one question this study raises is whether undergraduates have rich
opportunities to enhance their content knowledge of mathematics through verifying and analyzing propositions.
Even though reading, writing, and understanding complete proofs and counterexamples are fundamental activities in
advanced mathematics, a primary finding from this study shows that the majority of participants were not equipped with
adequate knowledge about producing mathematical proofs and counterexamples. Many students seemed to simply manipulate symbols, making logical errors and not exhibiting clear understandings of the statement and continuous functions,
which supports Harel and Sowder’s (1998) research on non-referential symbolic proof scheme. Merely applying symbolic
manipulations without reflecting on proof processes is not an effective way to promote mathematical reasoning and understanding, because being able to manipulate mathematical symbols in flexible ways is an important skill in proving theorems
and writing proofs (Weber, 2001, 2004). In order to help undergraduates gain deeper understandings of mathematical concepts through proving and refuting, mathematics instructors may consider a stronger emphasis on explaining mathematical
ideas related to symbols.
While a mathematical proof is “neither rigorous nor complete” (Hanna, 1991, p. 59), it should incorporate the important
ideas of sets of accepted statements, modes of argumentation, and modes of argument representation in the classroom community (Stylianides, 2007). Common sense suggests that satisfactory mathematical knowledge is a prerequisite for verifying the
truth and falsity of statements and producing proofs and counterexamples (Selden & Selden, 2009). We found that students
need both adequate knowledge of choosing accepted definitions, axioms, and facts and an understanding of what counts as
valid proof and counterexample, which supports Weber’s (2001) finding that students need strategic knowledge to construct
proofs. In order to assist undergraduates in writing complete formal proofs and counterexamples with and understanding
them, mathematics instructors should expose students to the same concepts twice to allow them to focus on proof and counterexample techniques (Sowder, 2004) and pay more attention to students’ conceptions as manifested in their proof and
counterexample work. Changing pedagogical processes can be difficult, but such an enhancement may be necessary to support students’ understanding of mathematical concepts and performance in writing proof and counterexamples. In addition
to changing teaching styles, we might also consider the implications for graduate students teaching advanced mathematics
courses. It is worth considering whether teaching assistants have satisfactory knowledge of proof and counterexample and
how their knowledge and pedagogies influence students’ learning.
Currently, few research studies have specifically focused on undergraduate mathematics majors’ abilities to produce
proofs for the statements they believe to be true and counterexamples for the statements they believe to be false in the
domain of continuous functions. Empirical studies already show that undergraduate students and mathematics teachers
have difficulty determining a given claim (Barkai et al., 2002; Riley, 2003); however, little is known about the kinds of
strategies students use to verify true and false propositions (Buchbinder & Zaslavsky, 2007). Further research is needed in
order to gain more insight into the strategies undergraduates use to determine the validity of statements and the relationship between undergraduates’ understandings of continuity and proofs and counterexamples. This research could involve
designing more mathematical statements, conducting intensive interviews with students, and observing their lectures and
discussion sections to understand their perspectives. In doing so, we will have a better understanding of how to support
undergraduate students to learn proof and counterexample meaningfully. This paper highlights the need to devote more
attention to empowering instructors in their teaching and undergraduates in their learning to verify given propositions with
complete proofs and counterexamples that demonstrate a deep understanding of concepts.
Acknowledgements
The authors would like to thank Dr. Erica Halverson, Beth Godbee, Clara Burke, and Caroline Williams, and anonymous
reviewers for their thoughtful comments on prior versions of this paper; Dr. Haw-Yaw Shy for helping to collect the data;
and Wan-Ping Wang for completing the reliability coding for this study.
Y.-Y. Ko, E. Knuth / Journal of Mathematical Behavior 28 (2009) 68–77
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