The rebirth of proof
in school mathematics in the United States
?
Eric Knuth
University of Wisconsin, USA
In the not too distant past, proof was expected to play a
significant role in the mathematics education of all
students in the United States. In fact, the hallmark of the
new math curriculum of the late fifties and early sixties
was an emphasis on "rigor in the presentation of
mathematical ideas and on rigorous proof in particular"
(Hanna, 1983, p. 1). The curriculum received, however, its
share of criticism regarding the proliferation of proofs in
the curriculum and their implementation in teachers'
instructional practices. In large part as a response to this
criticism, and in consideration of the effect the curriculum
had on students' (as well as teachers') conceptions of proof
(not to mention of mathematics), the 1989 Curriculum and
Evaluation Standards for School Mathematics (National
Council of Teachers of Mathematics [NCTM])
de-emphasized proof in school mathematics, choosing instead
to emphasize reasoning (J. Kilpatrick, personal
communication, March, 1999). As a consequence, students have
had limited encounters with proof in school mathematics and,
not surprisingly, have found the study of proof difficult
(e.g., Chazan, 1993; Sowder & Harel, 1998; Usiskin,
1987).
This absence of proof, however, did not go
unnoticed and, in fact, has been a target of criticism as
well. Wu (1996) argued that the scarcity of proof outside of
geometry is a glaring defect in the present-day mathematics
education in high school, namely, the fact that outside
geometry there are essentially no proofs. Even as anomalies
in education go, this is certainly more anomalous than
others inasmuch as it presents a totally falsified picture
of mathematics itself (p. 228).
Similarly, Schoenfeld (1994) suggested,
"Proof is not a thing separable from mathematics as it
appears to be in our curricula; it is an essential component
of doing, communicating, and recording mathematics" (p. 76).
Reflecting an awareness of such criticism, as well as
embracing the central role of proof in mathematics, recent
reform efforts in the United States are calling for
substantial changes in both school mathematics curricula and
teachers' instructional practices with respect to proof.
In contrast to the status of proof in the
previous national standards document (i.e., NCTM, 1989), its
position has been significantly elevated in the most recent
national standards document (NCTM, 1998)&emdash;a document
intended to guide the revision of school mathematics in the
United States into the next millennium. Not only has proof
been upgraded to an actual standard in this latest document,
Mathematical Reasoning and Proof, but it has also received a
much more prominent role throughout the entire school
mathematics curriculum and is expected to be a part of all
students' school mathematics experiences. In particular, the
Principles and Standards for School Mathematics (NCTM, 1998)
recommends that in grades pre-K to 12:
Mathematics instructional programs should focus
on learning to reason and construct proofs as part of
understanding mathematics so that all students --
- recognize reasoning and proof as
essential and powerful parts of mathematics;
- make and investigate mathematical
conjectures;
- develop and evaluate mathematical
arguments and proofs; [and]
- select and use various types of
reasoning and methods of proof as appropriate (p. 80).
It is certainly clear from reviewing these
recommendations that proof is expected to again play a
significant role in school mathematics in the United States.
Yet an important question&emdash;one that has serious
implications for the successful enactment of proof in school
mathematics&emdash;remains: Are teachers prepared to enact
these recommendations in their instructional practices?
Approaches designed to enhance the role of
proof in the classroom, and accordingly students'
conceptions of proof, require a tremendous amount of a
teacher (Chazan, 1990). Yet, mathematics teacher education
has, traditionally, not adequately prepared teachers to
successfully enact the lofty expectations set forth in
reform documents (Ross, 1998). This inadequate preparation
is particularly troublesome with respect to proof
considering the often limited conceptions of proof held by
many prospective teachers (e.g., Goetting, 1995; Harel &
Sowder, 1998; Jones, 1997; Martin & Harel, 1989; Simon
& Blume, 1996). Further, research has typically not
examined teachers' conceptions of proof as individuals who
are teachers of school mathematics; rather, research has
primarily focused on teachers' conceptions of proof as
individuals who are knowledgeable about mathematics.
In this article, I briefly discuss the
results of a study which was designed both to address the
aforementioned question and to identify areas of need for
preparing teachers to successfully enact the recommendations
of reform with respect to proof (see Knuth, 1999, 2000a,
2000b, for greater detail). In particular, I discuss the
conceptions of proof held by 18 experienced secondary school
mathematics teachers, focusing specifically on their
conceptions of proof as individuals who are teachers of
school mathematics.
The Role of Proof in Secondary
School Mathematics
Teachers suggested several roles for proof in secondary
school mathematics, two of which serve important educational
functions and, in addition, speak toward aspects of reform.
First, teachers suggested that proof plays a role in
answering why a statement is true. In this case, rather than
explain why a statement is true, proof serves to show how a
statement came to be true. For example, teachers viewed a
derivation of the quadratic formula as an illustrative
example of this role of proof&emdash;one could follow the
progression of steps in the derivation to understand how the
general formula was derived (i.e., "why" it was true). As
one teacher commented, "It gives a way for kids to
understand why things are the way they are….Instead of
just accepting [formulas] at face value, proofs give
[students] a way of justifying the formulas."
Second, and integrally related to the
first, teachers mentioned the role of proof in fostering
student autonomy. In order for students to be autonomous in
mathematics classrooms, they must be able to create their
own knowledge through validating their own as well as their
classmates' knowledge claims. One teacher suggested that
proof "allows your students to be independent thinkers,
instead of just robots who are told this is the
relationship, this is what works....Students don't have to
rely on a teacher or a book to give them information."
Again, this role is important pedagogically as it enables
students to become producers of knowledge rather than
consumers of other's knowledge. Further, this role of proof
parallels one of the major mathematical goals identified in
the latest Standards document: "A major goal of school
mathematics instructional programs should be to create
autonomous learners" (NCTM, 1998, p. 35).
Noticeably missing from among the roles
teachers suggested&emdash;and a role of proof many
mathematicians view as important (Hanna, 1983, 1990; Hersh,
1993)&emdash;was a recognition of proof serving an
explanatory capacity, that is, proof as a means of promoting
understanding of the underlying mathematics. In some
respects, it is not surprising that this role was not
mentioned by any of the teachers; for many teachers, the
focus of their previous experiences with proof as students
themselves was primarily on the deductive mechanism or on
the end result rather than on the underlying mathematical
relationships illuminated by a proof (e.g., Chazan, 1993;
Goetting, 1995; Harel & Sowder, 1998). Nevertheless, of
all the roles of proof, its role in promoting understanding
is, perhaps, the most significant from an educational
perspective. As Hersh (1993) suggested, "at the high-school
or undergraduate level, its primary role is explaining" (p.
398). Ross (1998) went as far as to suggest that "the
emphasis on proof [in school mathematics] should be
more on its educational value than on formal correctness.
Time need not be wasted on the technical details of proofs,
or even entire proofs, that do not lead to understanding or
insight" (p. 3).
Proof for All?
In contrast to the central role of proof in the
discipline of mathematics, the majority of teachers did not
consider proof to play a central role in secondary school
mathematics, questioning, in fact, its appropriateness for
all students. As one teacher commented: "[Proof is]
for kids who are going to be going into mathematics and
probably studying mathematics in college. Tenth grade and
under, I'm not convinced that proof has a real role with
them." Another teacher spoke more adamantly about the
appropriateness of proof for all students: "I think they're
[i.e., advocates of proof throughout school
mathematics] smoking crack. I'd like to see how that
would happen, what that looks like in a classroom." Thus for
these teachers, proof seems to be an important idea only for
those students enrolled in advanced mathematics classes and
for those students who will most likely be studying
mathematics in college. This perspective is in stark
contrast to the message being put forth by advocates of
reform, namely, that "reasoning and proof [italics
added] must be a consistent part of students'
mathematical experience in grades pre-K-12" (NCTM, 1998, p.
85).
Many teachers even more finely delineated
the role of proof in upper level mathematics courses,
relegating proof primarily to geometry. Moreover, even those
teachers who didn't specifically cite geometry as the home
of proof in secondary school mathematics, stated that its
presence in other upper level mathematics courses was
implicit at best, and absent at worse. As one teacher
suggested, "In secondary school mathematics proof is not a
big part of algebra courses or analysis courses." Again, a
view that is inconsistent with the message of reform,
"Formal proofs occur in all areas of mathematics and
students' school experience with proof should not be limited
to geometry" (NCTM, 1998, p. 316), and is one that does not
reflect the essence of proof in mathematics.
Teachers did, however, view informal
proofs (e.g., arguments based on empirical evidence) as
playing a significant role in the mathematics education of
all students. Experiences with more informal methods of
proof can provide students with opportunities to formulate
and investigate conjectures&emdash;both important aspects of
mathematical practice&emdash;and may help "students develop
an inner compulsion to understand why a conjecture is true"
(Hoyles, 1997, p. 8). Such practices are also reflective of
the process of experimentation in mathematics: "Most
mathematicians spend a lot of time thinking about and
analyzing particular examples. This motivates future
development of theory and gives a deeper understanding of
existing theory" (Epstein & Levy, 1995, p. 670). For
many teachers, informal proofs were viewed as often serving
this very function (in higher level classes and, in
particular, geometry), namely, as precursors to the
development of more formal methods of proof&emdash;to the
"development of theory." One teacher described this process
in one of her classes: "This [experimentation]
students do very early on to show that it works. Then when
we introduce other geometry concepts, we come back to this
and prove it formally."
For students in lower level mathematics
classes, however, their encounters with proof are limited to
informal proofs. As one teacher commented, "When they say 'I
noticed this pattern and I tested it out for quite a few
cases,' you tell them good job. For them, that's a proof.
You don't bother them with the general cases." In fact, few
teachers even discuss the limitations of such informal
proofs with their students; students are left to believe
that their informal arguments are indeed proofs. As Wu
(1996) noted, this emphasis on informal proof, even for
students in lower level mathematics classes, is "a move in
the right direction only if it is a supplement to, rather
than a replacement of, the teaching of correct mathematical
reasoning; that is, proofs" (p. 226).
In sum, it is evident that the reform
recommendation of "proof for all" is not a view most
teachers hold. Rather than providing all students with
opportunities for developing "an increasingly sophisticated
understanding of mathematical proof" (NCTM, 1998, p. 316)
and "an appreciation for the necessity and power of
mathematical proof for establishing the truth of their
conjectures" (p. 317), teachers tended to view these goals
as appropriate opportunities primarily for students enrolled
in higher level mathematics classes&emdash;the minority of
students who study mathematics in secondary school. Further,
even for those students who are enrolled in upper level
mathematics classes, the teachers' tended to see geometry as
the course where students explicitly encounter the practice
of proving. Thus, if these teachers are representative, then
for the majority of students&emdash;students enrolled in
lower level mathematics classes and students in upper level
mathematics classes outside of geometry&emdash;their
secondary school mathematics experiences will most likely
not include significant encounters with more formal methods
of proving.
Implications for Mathematics
Teacher Education
If teachers are to be successful integrating proof
throughout secondary school mathematics curricula, then
their conceptions of proof must be enhanced. The
responsibility for enhancing their conceptions of proof lies
with both mathematicians and mathematics educators, parties
who are responsible for the nature of teachers' experiences
with proof in their university mathematics classes and
teacher education classes respectively.
In preparing mathematics teachers to meet
the demands of reform, university mathematics professors
need to engage teachers in classroom experiences with proof
that are more reflective of proof in their own practices. As
Alibert and Thomas (1991) suggested,
[the] context in which students meet
proofs in mathematics may greatly influence their
perception of the value of proof. By establishing an
environment in which students may see and experience
first-hand what is necessary for them to convince others,
of the truth or falsehood of propositions, proof becomes
an instrument of personal value which they will be
happier to use [or teach] in the future (p. 230).
In short, teachers need to experience proof as a
meaningful tool for studying and communicating mathematics
rather than as an often meaningless exercise to be done for
the professor. Experiences of the former nature may then
influence the conceptions of proof teachers develop, which
in turn, influence the experiences with proof their students
encounter in secondary school mathematics classrooms.
Perhaps the greatest challenge facing
mathematics teacher educators is changing teachers' beliefs
about the appropriateness of proof for all students and in
all classes. A starting point may be to engage teachers in
discussions about what constitutes proof. Does what suffices
as proof in the discipline differ from what suffices as
proof in secondary school mathematics? Is the acceptance of
an argument as proof dependent on the particular community
of practice? Is a proof a proof or are there levels of
proof? If teachers have limited understandings of what
constitutes proof, then it's not surprising that they may
perceive proof as inappropriate for most secondary school
students. Having teachers construct and present proofs of
secondary school mathematics tasks&emdash;tasks from various
content areas and levels&emdash;provides a forum for
discussing expectations of proof for students at differing
levels of mathematical ability and in different mathematics
courses. In addition, providing teachers with opportunities
to discuss the pedagogical merits of different arguments for
a task in terms of an arguments' explanatory qualities may
engender in teachers a richer perspective regarding the
arguments they select for use in their own instruction (cf.
Hanna, 1990). Engaging teachers in all of the aforementioned
activities may result in their adopting a view of proof as a
tool for studying and understanding mathematics&emdash;an
appropriate goal for all students&emdash;rather than as a
topic of study&emdash;a perceived goal appropriate for a
minority of students.
Concluding Remarks
As Edwards (1997) suggested, "the teaching of proof that
takes place in many secondary level mathematics classrooms
has often been inconsistent with both the purpose and
practice of proving as carried out by established
mathematicians" (p. 187). In some sense this is not
surprising; secondary school mathematics teachers&emdash;as
well as their students&emdash;are, arguably, not
mathematicians. Yet, the nature of classroom mathematical
practices envisioned by recent mathematics education reform
initiatives, and which teachers are expected to establish,
reflects the essence of practice in the discipline (Hoyles,
1997). This vision of mathematical practice, however, places
serious demands on secondary school mathematics teachers,
and their success in responding to these demands depends
largely on their conceptions of proof. At the beginning of
this article, I posed the following question: Are school
mathematics teachers prepared to enact the current reform
recommendations regarding proof in their instructional
practices? In response, I suggest that the successful
enactment of such practices may be difficult for teachers.
It is my hope that the findings of this study (some of which
were briefly presented in this article) provide mathematics
educators with information needed to better prepare teachers
to successfully enact these new recommendations. I agree
wholeheartedly with Schoenfeld (1994), who concluded, "Do we
need proof in school mathematics? Absolutely! Need I say
more? Absolutely" (p. 74).
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©
Eric Knuth
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