Proof in Geometry
Running head: PROOF IN GEOMETRY
Research and Practice: Proof in the Geometry Classroom
Samuel Otten
Michigan State University
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Proof in Geometry
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Research and Practice: Proof in the Geometry Classroom
In the discipline of mathematics, very little is regarded as highly as proof. It is proof that
mathematicians work toward and work from. It is proof that sets mathematics apart from the
empirical sciences. Proof is the currency of professional mathematics—if you possess many then
you are well off; if you are lacking it will be a struggle to get by. In school mathematics, on the
other hand, proof was confined to a single course for nearly the entire twentieth century and
remains so confined in many schools today. The sole refuge of proof in school mathematics has
been high school geometry. If proof is of highest importance in the field of mathematics, why
has it been scarce to be found in the school curriculum?
Some may argue in the other direction by saying that even one course involving proof is
one course too many, since very few students will become mathematicians. A person following
this line of thought is likely focusing on the proven statements, which admittedly are of little
educational value and may never be used in daily life (Fawcett, 1938, p. 117; Polya, 1957, p.
216). However, the processes of thought which are cultivated by the learning of proof make
unique and important contributions to a student’s educational experience. For instance, a student
who works with proof and has some understanding of its nature is more likely to appreciate the
need for clear definitions, be able to evaluate alleged evidence, expose the assumptions on which
conclusions are based, and in general is more likely to reason soundly (Fawcett, 1938, p. 6;
Polya, 1957, p. 217). These abilities are clearly valuable in myriad aspects of life and are a
worthwhile goal for the education of any student.
The National Council of Teachers of Mathematics (NCTM, 1989) recognized the
discrepancy between the status of proof in the eyes of mathematicians versus proof in the eyes of
students, and the Council also saw the value that proof adds to an individual’s education. Thus
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they included a proof process standard in their call for mathematics education reform.
Specifically, NCTM called for proof and justification to become an explicit and pervasive part of
mathematics instruction, which meant incorporating proof into all areas of mathematics for
students of all ages. This call led to an effort by researchers and teachers to formulate a
conception of what proof looked like at various grade levels (e.g., Stylianides, 2007) and in
classes other than geometry (e.g., Olmstead, 2007; Otten, Herbel-Eisenmann, & Males, Preprint).
As the mathematics education community works toward giving proof a more prominent
and widespread place in the classroom, I believe it is important to look back upon the area of
mathematics where proof has been most often found—geometry. This reflection can serve
several purposes. First, we can review historical documents in an attempt to understand why
educators maintained proof in geometry over the years, since it would have been rather easy to
have excised it completely. This may shed light on the motivations of the current movement to
reinvigorate proof in the classroom. Second, we can analyze some of the successes and failures
of proof instruction in geometry, which may allow us to increase the chance of success (and
avoid the failures) when teaching proof in other subject areas. Third, we can get a sense of the
“state of the art,” so to speak. It seems pertinent to view a snapshot of proof in the present-day
geometry classroom so that we know where we stand as we take steps forward.
This literature review consists of two primary sections. The first is more historic and
research-based in nature and is meant to present a few significant works which provide us with
important information about proof in geometry. This will also provide a framework for the
remainder of the paper. The second section delves into the classroom as it brings teacher articles
as well as classroom-centered studies into dialogue with the literature from section one.
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Selections from History and Research
In a large-scale study, Senk (1985) found that of high school students who had taken a
full-year course in geometry which included proof, only 30% had reached a 75% mastery of
proof-writing. Furthermore, 25% had “virtually no competence in writing proofs” (p. 453) and
another 25% were only able to complete trivial proofs. Perhaps even more disturbing was the
finding that a significant number of students stated the theorem to be proved as the proof of the
statement, which points to a fundamental lack of understanding with regard to the logical nature
of proof. What led to the dismal state of things found in the mid-1980’s? A full answer to this
question is certainly beyond the scope of the present work, but we may find a partial answer by
looking to historic texts on the subject and other research articles.
Historical Perspective
The study of deductive geometry, until the mid-nineteenth century, was found only at the
college level (Fawcett, 1938). At that point the ever-increasing number of college course
requirements made it so that most students were unable to study deductive geometry.
Fortunately, high school was developing during this time period in such a way that it could
provide a home for geometry, but much of the deductive character was lost in the move. Later, in
the 1890’s, an educational reform movement took place which was characterized by the notion
that schools take responsibility not only for the presentation of facts and skills, but for the
general intellectual activity of the students (Herbst, 2002). For mathematics in particular this
meant the inclusion of proof in the curriculum, and the logical place for it was geometry due to
the fame of Euclid’s Elements—an axiomatic tour de force that laid out all the major topics of
elementary geometry in a proposition-proof format.
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So it was that proof found itself in the geometry corner of the high school curriculum,
and in order for proof to be accessible to all students the two-column proof format became the
norm throughout the twentieth century (Herbst, 2002). Unfortunately, instruction based around
the two-column format has a tendency to turn proof into a procedure or something to be
memorized (NCTM, 2000), which is the antithesis to the true nature of proof and does not lead to
the positive modes of thought that are the primary goal of proof instruction. (More negative
characteristics of the traditional, two-column approach will be addressed in the following
subsection.) Fawcett (1938) seemed to be aware of this problem when he wrote his classic work
on proof instruction.
Fawcett’s dissertation (which was published by NCTM as a yearbook) presents in great
detail an instructional strategy for demonstrative geometry based on the following assumptions:
1. Secondary students have the ability to reason and reason accurately before they
begin a course in demonstrative geometry.
2. Students should have opportunities to reason about material in their own way.
3. The logical processes guiding the work should be those of the students and not
those of the teacher.
4. Opportunities should be provided for the application of deductive reasoning to nonmathematical material.
This framework led to geometry classrooms characterized by student-generated texts, studentgenerated conjectures, and an emphasis on the method of proof rather than the statements being
proved. (It should be noted that the teacher maintains an amount of control over the direction of
the course by suggesting points of inquiry and by manipulating the situations of discovery.)
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When contrasted with a usual course in geometry, Fawcett found that his instructional
program produced students with an improvement in reflective thinking practices. Moreover,
these reflective tendencies were general in the sense that they transferred beyond the domain of
mathematics. For example, students reported analyzing the logic of advertisements, the
assumptions made in sermons and political speeches, as well as the validity of newspaper
editorials. The parents of the students, though they had a largely unfavorable attitude toward
geometry, also admitted that their children had begun to apply analytic reasoning in various
domains outside of school and acknowledged that the course had been valuable. What of the
specific geometric content? Was it lost in Fawcett’s outpouring of reasoning and reflection? The
results indicated that the subject matter was achieved to the same level under his program as in a
usual geometry course.
Alas, the promising results of Fawcett’s study did not unseat the two-column format from
its place of prominence in secondary geometry courses (Herbst, 2002). There were, however,
several other significant efforts made to improve the nature of mathematics education. Polya
(1957) attempted to infuse an explicit emphasis on problem solving and heuristics into
mathematical instruction. This included a strong component of geometric proof. Polya wrote that
if a student “failed to get acquainted with geometric proofs, he missed the best and simplest
examples of true evidence and he missed the best opportunity to acquire the idea of strict
reasoning” (pp. 216-217). Here Polya is referring to a notion of proof as a form of problem
solving, not as a memorization of facts and procedures.
Additionally, the vast body of work of Piaget contained a significant theory on the
development of proof skills in children (Clements & Battista, 1992; Pandiscio & Orton, 1998).
Prior to the age of 8 or 9, observations and individual conclusions are not integrated into a
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coherent system of thought. Hence the reasoning is often self-contradictory and illogical. From
the approximate ages of 8 to 12, students begin to make reasoned predictions and may attempt to
justify their thoughts. However, the justification usually relies entirely on empirical data or some
form of inductive reasoning. Beyond age 12, students are capable of deductive reasoning based
on assumptions and are thus prepared to operate within a mathematical system. Piaget posits that
progress through the levels is spurred on by the interaction of one individual’s thoughts with
another’s (Clements & Battista, 1992, p. 440). Under this theory, requiring the memorization of
definitions and proofs would do little to promote the thinking of students, but rather a collective
inquiry and discovery approach like that of Fawcett would be more likely to bring about the
reasoned contact with others which leads to formal deductive abilities.
Another attempt to improve the state of proof in math education can be found in the work
of van Hiele (e.g., van Hiele, 1984). He and his wife conceptualized different levels of student
geometric reasoning. The base level (Level 0) is characterized by students reasoning about
shapes strictly as a whole. A student reasoning in the first level (Level 1) informally analyzes
parts and attributes of shapes. Level 2 reasoning involves ordering the properties of concepts and
the ability to distinguish between necessary and sufficient conditions. Level 3, referred to as
deduction, is characterized by students’ reasoning within a mathematical system including
undefined terms, axioms, definitions, and theorems. Finally, a student reasoning at the fourth
level (Level 4) is able to compare and contrast different geometric systems and work within a
particular geometry without a model.
The work of van Hiele also included a theory of instruction designed to help students
move through the levels of geometric reasoning. The process begins by engaging students in
inquiry so they may discover structures in the material being learned. This is followed by
Proof in Geometry
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directed orientation in which the teacher presents material in such a way that the characteristic
structure is revealed gradually. Next is explication in which the student connects language and
symbols to the material they have been experiencing. The penultimate phase is that of free
orientation wherein students can work through fairly sophisticated tasks because they are now
quite familiar with the material. Finally, it is during integration that the teacher leads students to
survey and organize the material and relations they have been exploring throughout the prior
phases. Ideally, this progression culminates in the achievement of the next van Hiele level of
reasoning, and then starts anew for the following level.
With regard to proof, the van Hiele theory suggests that students need to be reasoning in
the third or fourth level in order to be successful in a deductive geometry course. Thus they need
to have prior experiences working through the first and second levels; that is to say, it would be
unfair to expect students to be successful in a deductive setting unless they have had
opportunities to reason about and analyze the components of figures as well as opportunities to
consider necessary and sufficient conditions and reason inductively. This is similar to the proof
framework of Sowder and Harel (1998) in which a student’s experience in a transformational
proof scheme (i.e., justifications based on reasoned consideration of a general case) is a
necessary prerequisite to the axiomatic proof scheme. Therefore, let us examine the research
with these thoughts in mind.
Research Literature
We have seen above the results of a study on the general success (or lack thereof) that
students achieve in writing proofs (Senk, 1985), and this was indicative of some of the problems
with the traditional manner in which proof was taught in high school geometry courses. We then
surveyed a few theoretical frameworks which illuminated conceptually some of the deficiencies
Proof in Geometry
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that the two-column proof format, for instance, entails. Senk also conducted a study that
attempted to make an explicit connection between one of the theories—that of van Hiele—with
the performance of students in proof-writing (1989). She found that the van Hiele levels of
students, as measured by a multiple-choice test, were significantly predictive of success in
generating geometric proofs. Her results support van Hiele’s characterization of Level 3 as
deductive in nature, with the caveat that students reasoning at Level 2 were not entirely unable to
generate proofs. Though several questions can be raised about her method (e.g., the strong
assumption of linearity in the van Hiele levels), it is difficult to argue with her conclusion that
incoming knowledge has an important impact on the success of students in high school
geometry.
Shaughnessy and Burger (1985) made a more significant connection between the van
Hiele levels and proof in geometry. First, they noted that miscommunication often occurred
because students were reasoning at different levels than the teacher, so perceptions and the use of
language were different. Second, the students had had insufficient opportunities to develop their
sense of necessary and sufficient conditions (Level 2) which led to difficulty when they were
thrust into an axiomatic system. Indeed, most students at that time entered their high school
geometry course reasoning at Level 0 or Level 1, but to have a good chance of success they
should have come in at Level 2. Thus Shaughnessy and Burger called for an increase in the
teaching of informal geometry to high school students to better prepare students for success in an
axiomatic proof environment. Overall, their research (as well as Senk’s) pointed to the
usefulness of the van Hiele framework for interpreting student reasoning in geometry.
Knuth and Elliot (1998) investigated students’ conception of proof under what could be
considered a Piagetian perspective (though they did not identify this explicitly). They placed
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emphasis on “mathematical reasoning through the social interactions occurring within the
classroom community” (p. 714) which is aligned with the impetus that Piaget identified as
moving students through the stages of proof reasoning. In response to the task in their study,
Knuth and Elliot found that the majority of claims made by students were based on empirical
evidence, even those made by students who would be considered mathematically sophisticated.
In other words, the students had not yet reached Piaget’s deductive stage. Knuth and Elliot
conclude that it is unlikely this progression will occur as long as teachers reason based on
examples and do not cultivate a culture of proof in the classroom in which argumentation and
convincing take place.
In addition to those of van Hiele and Piaget, there has also been research supporting
Fawcett’s conception of proof in geometry, specifically with regard to the inadequacy of the
traditional approach. For instance, Brumfield (1973, cited in Clements & Battista, 1992) found
that more than 80% of students who had taken a traditional geometry course were unable to list a
single postulate, and 40% were unable to list a single theorem. Schoenfeld (1986) and Chazan
(1993) uncovered a significant disconnect in students’ minds between deduction and empirical
investigation; that is, students viewed empiricism as the means for determining the truth of a
statement and deduction as an arbitrary exercise required by math teachers and textbooks.
Ironically, the students saw no justification for or from proof. There is still more that points to
what Fawcett foresaw as the danger of a traditional approach. Even after completing a course in
axiomatic geometry, students often accept incorrect arguments as valid, believe that checks are
still needed after a statement is proven, and maintain that one counterexample is not sufficient to
disprove a claim (Clements & Battista, 1992). Moreover, students do not appreciate the full
function of proof as a means of verification, illumination, and systematization (Clements, 2003).
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In short, the research suggests that traditional instruction fails to instill in students an
understanding of the nature of proof.
Carroll (1977) took a slightly different approach in that he identified differences within
traditional approaches to geometric proof rather than grouping them together into a single
category. He noted that proof can be presented in a synthetic, analytic or combination manner
and set out to identify the optimal strategy. The synthetic approach involves starting with a
hypothesis and reasoning deductively to a conclusion. The analytic approach reverses this by
starting with a conclusion and then forming a chain of reasoning back to the hypothesis. The
combination approach mixes these two. Carroll found that presenting proof analytically was the
weakest instructional strategy of the three, especially in terms of dealing with extraneous
information in the hypothesis. While it is important to recognize Carroll’s point that traditional
instruction is not homogenous, the results of his study should not be given too much emphasis.
He enacted instructional conditions for only six days, and according to the research cited above,
this likely took place in an environment where proof was largely misunderstood and
unappreciated by the students. Furthermore, it is very probable that optimal strategies for proof
instruction lie somewhere off Carroll’s list.
In summary, research suggests that instruction based on two-column proof and other
formats that teach proof as a finished, rigorous product have generally failed. This was true
regardless of which theoretical frame the researchers used. Such approaches lead students to
believe that proof is an exercise in logic that validates unimportant statements (Herbst, 2002), or
that proof is a forced school task to verify something the students are already convinced of based
on examples (de Villiers, 1995). Thus, it seems that students must be included in the process of
inquiry, investigation, and discovery so that they may see firsthand the nature of conjecture and
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proof (Hanna, 1989). The light at the end of the tunnel is that, in addition to Fawcett’s study,
there are results suggesting that improvement can be made with regard to proof in geometry. For
example, Clements (2003) suggested that efforts based on a cognitive model of conjecturing and
argumentation may be more successful than simply introducing more informal geometry earlier,
and Greeno and Magone (reported in Driscoll, 1983) found that a short period of training in the
nature of proof led to improved proof checking and proof construction by students. Senk also
made the optimistic point that “much of a student’s achievement in writing geometry proofs is
due to factors within the direct control of the teacher and the curriculum” (Senk, 1989, p. 319).
Therefore, let us glimpse classroom practice with regard to proof in geometry.
Selections from the Classroom
Practice Connected to History and Research
Clements and Battista (1995) encapsulated much of what was discussed above when they
wrote, “Research suggests that alternatives to axiomatic approaches can be successful in moving
students toward meaningful justifications of ideas…In these approaches, students worked
cooperatively, making conjectures, resolving conflicts by presenting arguments and presenting
arguments, proving nonobvious statements, and formulating hypotheses to prove” (pp. 50-51).
They connected this to practice by encouraging teachers to actively involve students in rich
mathematical discourse and discovery. Moreover, they suggested that visual justification and
empirical reasoning be allowed in the classroom as a basis for higher levels of reasoning. These
higher levels can then be achieved through teacher encouragement of justification and a gradual
illumination of the shortcomings of empiricism.
Clements and Battista, in the same article, presented two examples of such an
instructional approach. The first dealt with properties of similarity, which could be explored first
Proof in Geometry
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by paper-and-pencil or software-based enactments of dilations. This could be succeeded by
investigations into the properties of the figures, a discussion of various definitions of similarity,
and finally deductive work concerning propositions of similarity. The second example concerned
cyclic quadrilaterals. Clements and Battista illuminated this geometric situation as ripe with
possible conjectures which could be discovered empirically and then proven deductively.
It is clear that one of the primary aspects of the approach of Clements and Battista (as
well as others below) is that empirical and inductive reasoning be allowed, even promoted, to
then be followed by deductive, more rigorous mathematical reasoning. This hinges on students
eventually recognizing the limitations of empirical justification and argumentation. But as de
Villiers (1995) pointed out, this will not happen automatically since students are often and easily
convinced by a few examples. To assist the progression to advanced reasoning, Sultan (2007)
published an article equipping teachers with mathematical phenomena which lend themselves to
conjectures or arguments that turn out to be false, thus undermining the students’ reliance on
diagrams, examples, and so forth, and emphasizing the need for proof. One of Sultan’s examples
was a diagram which seemed to demonstrate that two perpendicular lines exist from a segment to
a point not on the segment (see figure 1).
This and other false proofs led to careful
deductions and discussions about the
nature of mathematical argumentation in
Sultan’s classes, and could do the same
in other classes to aide the transition
from empiricism to mathematical
deduction.
Figure 1: Angles ACP and BDP inscribe diameters
and thus are right, right? (Sultan, 2007)
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The preceding classroom incorporations of inductive and deductive methods, with a
specific emphasis on laying an empirical foundation for reasoning, are in agreement with van
Hiele’s theory of instruction. The teaching approach of Stallings-Roberts (1994) is also grounded
on van Hiele’s framework, and is simultaneously reminiscent of Fawcett’s classroom structure.
An integral component of her instruction is the physical construction of polygons, polyhedra, and
other geometric figures using manipulatives. This gives students the opportunity to develop their
Level 2 reasoning before being expected to function at van Hiele’s Level 3, which is more likely
to promote success (Shaughnessy & Burger, 1985). In addition, Stallings-Roberts did not issue
her students textbooks, but instead worked with them throughout the course to generate their
own text and their own axiomatic system. This led to meaningful discussions about definitions,
the need for undefined terms, and the nature of proof. Rather than the memorization and
procedure, which we saw from the research is characteristic of two-column proof, proof in the
classroom of Stallings-Roberts (as in Fawcett’s) became “a natural result of building and
recording an axiomatic system” (p. 406).
McGivney and DeFranco (1995) wrote an article based on their teaching practices which
fell explicitly under Polya’s framework of proof as problem solving. The example they presented
was a proposition to be proved in a high school
geometry class (see figure 2), but they
demonstrated how a teacher might go about
eliciting the proof through leading questions,
Figure 2: AF=GC, HF=HG, and DH=HE.
Prove that AB = BC. (McGivney &
DeFranco, 1995)
thus avoiding the two-column approach. In the
classroom vignette, the teacher prompted the
students to think of a similar problem they had
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solved, to identify the goal and possible subgoals, and to conduct a means-end analysis after
certain subgoals were achieved. The point that the authors raised was that heuristic strategies
could often be fruitful in developing a geometric proof and also promoted in students desirable
types of mathematical reasoning.
McGivney and DeFranco also reiterated findings of Schoenfeld—that students’ beliefs
about the nature of mathematics, and subsequently the nature of proof, are determined by “daily
practices and rituals of the classroom” (1995, p. 555). Thus, it is only natural that praise for
algorithmic solutions leads to a belief that algorithms are prized in mathematics. And if students
are expected to value the rigor and beauty of mathematics, then analyzing, conjecturing,
exploring, and proving should be included in their daily classroom experiences. A similar tone
was struck in the work of de Groot (2001) as he illustrated the fact that “student-to-student
discourse and careful teacher modeling support a transition path to more formal mathematical
reasoning” (p. 244). One example de Groot presented concerned the classification of
quadrilaterals and the notion of a rectangle as a parallelogram with at least one right angle. This
produced in a particular student a mental image that seemed impossible, and she declared “I want
to see such a rectangle!” This occurred in a middle school classroom, so it would not have been
appropriate (or successful) for the teacher to delve into a proof based on formal definitions and
parallel lines. Instead, the teacher prompted a transformational approach based on folding and
matching angles which led to a proof-like argument that was accessible to the class. In another
instance, de Groot highlighted a rich classroom dialogue in which students debated and reached
consensus on the concept of an arc of a circle. Overall, the various classroom episodes indicated
ways in which the classroom discourse could lay a foundation for mathematical proof and
reasoning.
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A Brief Look at Technology
The development of a variety of geometry software and their implications for the
teaching of geometric proof has a large presence in the existing literature. However, adequate
coverage of this topic would require an entire literature review itself, so in the current work only
a select few articles related to the incorporation of The Geometer’s Sketchpad (Jackiw, 1995)
into the teaching of geometry will be addressed.
Geometer’s Sketchpad, with its constructive and dynamical nature, offers rich
instructional possibilities (Clements, 2003). Giamati (1995) viewed the software as an
exploratory tool ideal for uncovering geometric invariants and testing conjectures. After sharing
a classroom experience, he concluded that “the power of The Geometer’s Sketchpad combined
with the power of proof gave a complete
illustration of the theorem involved and the
aspects of doing mathematics” (p. 458). The
exploration he was referring to involved
determining the center of rotation given two
congruent triangles. Using Sketchpad, students
were able to construct perpendicular bisectors of
the segments between corresponding points and
observe that they are concurrent at a point. They
Figure 3: Sketchpad was used to
quickly conjectured that this was the center of
determine the center of rotation.
rotation. Giamati used this as a launching point
(Giamati, 1995)
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into a proof of the conjecture. He then followed this with a discussion of the converse, and with
Sketchpad in hand, the students were able to construct a counterexample and demonstrate that
the conjecture was not biconditional.
Similarly, Izen (1998) presented his use of Sketchpad while working with his class on the
proposition that the angle bisector of an angle in a triangle divides the opposite side in a way that
is proportional to the other two sides of the triangle (see figure 4). To directly work toward the
proof of this theorem, Izen felt, would have been
beyond the reach of his class. However, by first
exploring the situation with Sketchpad, he was able
to successfully guide his students to a communityFigure 4: AD bisects angle BAC. Prove
that BD/DC=BA/AC. (Izen, 1998)
generated proof. This is representative of Izen’s
general teaching approach to geometry, in which he
provides opportunities for empirical exploration before later presenting or generating a proof
with the students. The resulting comprehension “leads to student’s ownership of the material and
prevents the student from feeling that the teacher is force-feeding information that makes no
sense” (p. 718). This refreshingly captures in practice much of the research that was presented
above.
Conclusion
Proof in high school geometry classes has traditionally been presented in a refined,
axiomatic form with a heavy reliance on two-column proofs (Herbst, 2002). Instructional
strategies of this type have been largely unsuccessful (e.g., Senk, 1985; Brumfield, 1973), and
often led students to view proof as procedural and memorization-based rather than reasoned and
motivated by understanding (e.g., Schoenfeld, 1986; Chazan, 1993; Knuth & Elliott, 1998).
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Several theoretical frameworks exist which are useful for the purpose of examining and
interpreting student reasoning with regard to proof in geometry, as well as for guiding instruction
(e.g., Fawcett, 1938; van Hiele, 1984; Polya, 1957). There is evidence that the movement toward
reform, as articulated by NCTM, is having a positive impact (e.g., Clements & Battista, 1992)
and is successfully making its way into classroom practice (e.g., Clements & Battista, 1995;
McGivney & DeFranco, 1995; Stallings-Roberts, 1994). In particular, the use of dynamic
geometry software appears to provide a useful means of enacting the reform (e.g., Giamati, 1995;
Izen, 1998).
As the mathematics education community works toward an incorporation of proof into all
subject areas and all grade levels, it is imperative that we consider what is to be learned from the
existing literature on proof in geometry. As I see it, the main point to be found in this literature
review harkens all the way back to Fawcett – that proof in mathematics is a rich and wonderful
process consisting of exploration, discovery, conjecture, induction, empiricism, argumentation,
reflection, refinement of thought, problem solving, and deduction, and the ideal way to teach
proof is to include the students fully in all of its aspects.
Proof in Geometry
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