Developing Student Understanding: The Case of Proof by Contradiction
Darryl Chamberlain Jr.
Georgia State University
Draga Vidakovic
Georgia State University
Proof is central to the curriculum for undergraduate mathematics majors. Despite transition-toproof courses designed to facilitate the transition from computation-based mathematics to proofbased mathematics, students continue to struggle with mathematical proof. In particular,
research suggests that proof by contradiction is a difficult proof methods for students to
construct and comprehend. The purpose of this paper is to discuss preliminary results on student
comprehension of proof by contradiction within a transition-to-proof course. Grounded in APOS
Theory, this paper will illustrate that students’ ability to negate quantification plays an early
role in student comprehension of proof by contradiction.
Key words: Proof by Contradiction, Transition-to-proof course, Teaching Experiment
Proof is central to the curriculum for undergraduate mathematics majors. Despite transitionto-proof courses designed to facilitate the transition from computation-based mathematics to
proof-based mathematics, students continue to struggle with mathematical proof (Samkoff &
Weber, 2015). In particular, research suggests that proof by contradiction is a difficult proof
methods for students to construct and comprehend (Antonini & Mariotti, 2008; Brown, 2011;
Harel & Sowder, 1998). The purpose of this paper is to discuss preliminary results on student
comprehension of proof by contradiction within a transition-to-proof course. In particular, this
paper will address the following question: How does a student's conception of quantification
affect their comprehension of proof by contradiction? Grounded in APOS Theory, a teaching
experiment was designed to assess the development of student understanding of proof by
contradiction. Results from a case study consisting of three teaching sessions with one student
will be presented and discussed, concluding with a discussion of how quantification negation
affects student comprehension of proof by contradiction.
APOS Theory and the ACE Teaching Cycle
APOS Theory is a cognitive framework that considers mathematical concepts to be
composed of mental Actions, Processes, and Objects that are organized into Schemas. An Action
is a transformation of Objects by the individual requiring memorized or external, step-by-step
instructions on how to perform the operation. As an individual reflects on an Action, he/she can
think of these Actions in his/her head without the need to actually perform them based on some
memorized facts or external guide; this is referred to as a Process. As an individual reflects on a
Process, they may think of the Process as a totality and can now perform transformations on the
Process; this totality is referred to as an Object. Finally, a Schema is an individual’s collection of
Actions, Processes, Objects, and other Schemas that are linked by some general principals to
form a coherent framework in the individual’s mind (Dubinsky & McDonald, 2001). Utilizing
the mental constructs of Actions, Processes, Objects, and Schemas, an outline of the hypothetical
constructions students may need to make in order to understand a concept can be developed,
referred to as a genetic decomposition. This genetic decomposition is used as a foundation to
develop instructional materials.
One such pedagogical approach aligned with APOS Theory is the ACE teaching cycle; an
instructional approach that consists of three phases: Activities, Classroom discussion, and
Exercises. In the Activities phase, students work in groups to complete tasks designed to promote
reflective abstraction rather than correct answers. These tasks should assist students in making
the mental constructions suggested by a genetic decomposition. In the Classroom discussion
phase, the instructor leads a discussion about mathematical concepts that the activities focused
on. Students take a prominent role in this discussion while the instructor guides the conversation
and presents an overview of what the students have discussed and introduces a formal,
mathematical way of presenting the concept. In the Exercises phase, students work on standard
problems designed to reinforce the Classroom discussion and support the continued development
of the mental constructions suggested by the genetic decomposition. The Exercises also provide
students with the opportunity to apply what they have learned in the Activities and Classroom
discussion phases to related mathematical concepts (Arnon et al., 2014).
Methodology
This paper is situated in a larger research project on how students develop an understanding
of proof by contradiction within a transition-to-proof course at a public R1 university in the
southeastern United States. Data for this preliminary report consists of three sessions of a
teaching experiment with a single student, Chandler, during summer 2016 from Bridge to Higher
Mathematics - the first course in which students are formally introduced to mathematical proofs
and their accompanying methods of proof. Chandler’s understanding is similar to 5 other
participants and can be considered representative of a general participant’s understanding. Unlike
a typical instructional sequence of ACE teaching cycle that, in a regular classroom, usually lasts
for a week, this teaching experiment consisted of 5 shorter, consecutive teaching sessions each
mimicking the ACE teaching cycle. That is, each session consisted of: students working on the
Activity worksheet focusing on a particular component of the genetic decomposition for proof by
contradiction (A); A discussion about the concepts from the worksheet (C); and a typical series
of proof comprehension questions (Exercises, E) related to the content of the worksheet. A more
detailed description for each session, including the role of the interviewer during each phase, is
described below. In addition, Table 1 provides an overview of the content for the first three
sessions with Chandler.
 Activity – Sessions began with a presented statement and proof. The interviewee would
talk out how this statement and proof can be converted to propositional logic. During this
phase, the interviewer acted as another student with incomplete knowledge.
 Classroom Discussion – After the Activity, the interviewer would ask the interviewee to
summarize the structure of the proof. At this point, a formal structure of the proof would
be given to the student, after which the interviewee would discuss how this structure
compared to the one written during the Activity. During this phase, the interviewer acted
as a knowledgeable agent who guided the student to make comparisons that would
develop student understanding.
 Exercise – After the Classroom Discussion phase, the interviewee answered
comprehension questions on their own, after which the interviewer prompted interviewee
for their answers and their thinking behind the answers provided. Note that as textbooks

do not normally provide comprehension questions on proofs, the proof comprehension
assessment model by Mejia-Ramos et al. (2012) was used to develop standard proof
comprehension questions for the Exercises. During this phase, the interviewer acted as a
knowledgeable agent to gain insight into the interviewee’s thinking.
Student’s Questions – After the Exercise phase, the interviewee was encouraged to ask
questions on any topic (not just those discussed in the interview).
Table 1. Overview of content per teaching experiment session
Activity
Session Converting
1
statements using
propositional logic.
Session Logical structure of
2
the statement “If
every even natural
number greater than
2 is the sum of two
primes, then every
odd natural number
greater than 5 is the
sum of three
primes”
Session Logical structure of
3
the statement
“There is no odd
integer than can be
expressed in the
form 4𝑗 − 1 and in
the form 4𝑘 + 1 for
integers 𝑗 and 𝑘.”
Classroom
Discussion
Discussion of
these
conversions and
quantification in
general.
Discussion of
procedure for
proof by
contradiction of
implication
statements.
Discussion of
procedure for
proof by
contradiction of
nonexistence
statements.
Exercise
Student’s
Questions
Comprehension
How do you
questions on a proof
negate the
of the statement “The statement
set of primes is
“(∀𝑥)(∀𝑦)
infinite”
((𝑃(𝑥) ∧ 𝑃(𝑦))
→ 𝑥 = 𝑦)”?
Comprehension
How do you write
questions on the
a proof by
proof of the statement contradiction for
“If every even natural the statement
number greater than 2 “𝑃 → 𝑄1 ∧ 𝑄2”
is the sum of two
and similar
primes, then every
statements?
odd natural number
greater than 5 is the
sum of three primes”
Comprehension
How do you write
questions on a proof
a proof for the
of the statement
Triangle
“There is no odd
Inequality?
integer than can be
expressed in the form
4𝑗 − 1 and in the
form 4𝑘 + 1 for
integers 𝑗 and 𝑘.”
Data Analysis
All sessions were video recorded and then transcribed by the interviewer. Transcripts of the
three sessions went through multiple passes of analysis. First, sections of the transcripts were
grouped by Activity, Classroom Discussion, Exercises, and Student’s Questions for each session.
Then, grouped sections were read together and general comments were made to find themes (if
any) outside of the genetic decomposition. Two themes emerged from these general comments:
(1) negating quantified statements and (2) procedures for proof writing. These themes were then
coded throughout all three sessions. Finally, each interview was coded for evidence of
developing an understanding of proof by contradiction. Due to space constraints, the rest of this
section will briefly discuss how Chandler negated quantified statements and how this affected his
comprehension of proof by contradiction.
Negating a statement is the first step toward writing a proof by contradiction and thus a
student’s conception of quantification negation directly affects proof comprehension of proofs by
contradiction. On several occasions, Chandler expressed the need for explicit rules in order to
negate quantified statements, indicating an Action conception of quantification negation. For
example, during Session 2 Chandler discussed having a table with instructions on how to negate
statements with single and multiple quantifiers, saying “Negation… I wish there was a table on
how to negate things and what they look like, I don’t know. There is this [points to page in
textbook with rules for negating ‘for all’ and ‘there exists’ statements] for quantifiers.” The
interviewer then found a table online with negations of logical operators that Chandler confirmed
was what he was looking for in addition to the page in the textbook.
In addition, Chandler had difficulties negating statements without first translating the
statement into propositional logic and (sometimes with prompting) using rules to negate the
propositional logic, again indicating an Action conception of quantification negation. For
example, in Session 2 Chandler was asked to give the negation of the statement “Every even
natural number greater than 2 is the sum of two primes.” An excerpt of what transpired is
provided below:
CHANDLER: Every even natural number greater than 2 is not the sum of two primes? No,
all? I don’t know.
[Interviewer prompts Chandler to write the statement in propositional logic]
INTERVIEWER: Alright. So for every n here, that’s how you said it to me. So parentheses
for all n, in the natural numbers, if n is greater than 2, then n is p + q. So what would be the
negation of this statement?
CHANDLER: There is a natural number…
INTERVIEWER: There is a natural number.
CHANDLER: n greater than 2 that is not equal to p + q.
Note that at first, Chandler was unable to state the negation of the claim. However, once the
claim was written in propositional logic, Chandler was able to recognize the negation of a “for
all” statement. Chandler’s need to convert the statement into propositional logic can be explained
by his reliance on external, specific rules for negating quantifiers.
In terms of proof by contradiction comprehension, a student with an Action conception of
quantification negation would need to have the logical structure of the proof explicitly written
out. If the initial logical step is omitted (as they sometimes are in textbooks), he or she would not
understand the proof as a whole. Chandler expressed a difficulty with an omitted negation in
Session 1 with the following claim and first line of a proof:
Claim: If every even natural number greater than 2 is the sum of two primes, then every odd
natural number greater than 5 is the sum of three primes.
Proof: Assume that every even natural number greater than 2 is the sum or two primes and
that it is not the case that every odd natural number greater than 5 is the sum of three primes.
Chandler is initially unable to recognize the hidden negation step of the above proof, stating “I’m
not sure when something contradicts, if you begin with 1, step 1. The assumption, or the given,
was that P and Q. If you begin with P and not Q, it’s already a contradiction, isn’t it? No?” Once
the intermediary step, “Assume it is not true that if every even natural number greater than 2 is
the sum of two primes, then every odd natural number greater than 5 is the sum of three primes”
and how this statement relates to the first statement of the proof “That is, assume that every
even…” was presented, Chandler indicated he understood the first statement was a rewrite of the
negation of the claim. This enabled Chandler to understand the logical underpinning of the proof.
Discussion
When Chandler was prompted to convert statements to propositional logic to negate and then
convert back to mathematical language, he was successful in understanding (locally and
globally) how statements of the proof logically followed. Converting statements to propositional
logic for negation also aided Chandler in understanding the “logical leaps” of presented proofs,
such as those that begin with the negated version of a statement and not with a phrase such as
“Assume the statement is not true; that is …” that would indicate a proof by contradiction. While
other authors have found difficulties negating quantification to affect student construction of
proofs (Antonini & Mariotti, 2008; Lin, Lee, & Wu Yu, 2003), Chandler’s comprehension after
logical quantification suggests that a student’s ability to negate quantification plays an early role
in student comprehension of proof by contradiction.
Future Plans
Data collection and analysis will continue in spring 2017. A major goal for this semester will
be to refine the preliminary genetic decomposition for proof by contradiction that guides the
design of this study, provided below.
Preliminary Genetic Decomposition for Proof by Contradiction
1. Students outline the propositional logic of a given proof to develop specific step-by-step
instructions to construct proofs by contradiction for the following types of statements: (i)
implication, (ii) single-level quantification, and (iii) property claim.
2. Students interiorize each of the Actions in Step 1 individually by examining the purpose of
statements of given proofs. These Processes become general steps to writing a proof by
contradiction for statements of the form (i), (ii), and (iii).
3. Students coordinate the Processes from Step 2 by comparing and contrasting the general
steps to determine the necessary steps for any proof by contradiction. This Process
becomes general steps to writing any proof by contradiction and identifying a proof as a
proof by contradiction.
4. Students encapsulate the Process in Step 3 as an Object by utilizing the law of excluded
middles to show proof by contradiction is a valid proof method. Students can now
comprehend proofs on a holistic level.
5. When necessary, students de-encapsulate the Object in Step 5 into a Process similar to Step
3 that then coordinates with a Process conception of quantification to prove multi-level
quantified statements.
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