Proof in Geometry

1

Running head: PROOF IN GEOMETRY

Research and Practice: Proof in the Geometry Classroom

Samuel Otten

Michigan State University

Proof in Geometry 2

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

Proof in Geometry 3

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.

Proof in Geometry 4

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.

Proof in Geometry 5

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 non-

mathematical material.

This framework led to geometry classrooms characterized by student-generated texts, student-

generated 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.)

Proof in Geometry 6

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

Proof in Geometry 7

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 8

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 9

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

Proof in Geometry 10

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).

Proof in Geometry 11

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

Proof in Geometry 12

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

Figure 1: Angles ACP and BDP inscribe diameters

and thus are right, right? (Sultan, 2007)

Proof in Geometry 13

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 2: AF=GC, HF=HG, and DH=HE.

Prove that AB = BC. (McGivney &

DeFranco, 1995)

Proof in Geometry 14

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,

thus avoiding the two-column approach. In the

classroom vignette, the teacher prompted the

students to think of a similar problem they had

Proof in Geometry 15

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.

Figure 3: Sketchpad was used to

determine the center of rotation.

(Giamati, 1995)

Proof in Geometry 16

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

quickly conjectured that this was the center of

rotation. Giamati used this as a launching point

into a proof of the conjecture. He then followed this with a discussion of the converse, and with

Figure 4: AD bisects angle BAC. Prove

that BD/DC=BA/AC. (Izen, 1998)

Proof in Geometry 17

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 community-

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).

Several theoretical frameworks exist which are useful for the purpose of examining and

Proof in Geometry 18

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.

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