**
Proof in Discovering Geometry pdf*** *

Geometry
has traditionally been the course in which a deductive structure is first
taught and formalized. This is tradition, but not good pedagogy. In a subject
that is highly visual we ask students to deny what they see until they have
performed this mysterious ritual called a two-column proof.

**The van Hiele Model **

Whenever
there is discussion about proof in geometry we should first review the research
of Dina van Hiele-Geldorf and her husband Pierre van Hiele. Their research
resulted in the van Hiele model of geometric reasoning. Research by numerous
mathematics educators including Usiskin 1982; Senk 1985; Burger and Shaughnessy
1986; Geddes and Fry 1988; Clements and Battista 1992; and more recently
Battista 2007, support the accuracy of the van Hiele model. The model consists
of five levels of geometric reasoning that students pass through from pre-
deductive visual thinking to an understanding of formal proof and mathematical
structures.

Level 0: *Visualization*
–students can identify rectangles by sight but squares are squares and are not
seen as special rectangles.

Level 1: *Analysis*
(descriptive) – students can identify properties of rectangles (by drawing,
measuring, and making models) but cannot yet derive other properties from
given.

Level 2: *Informal
Deduction* – students can give an informal argument to justify that the
figure is a rectangle from given properties.

Level 3: *Formal
Deduction* – students are capable of creating original logical
arguments.

Level 4: *Rigor*
– students are capable of reasoning about mathematical structures (i.e.
Euclidean -vs.- non-Euclidean geometries).

The
van Hiele model asserts these levels are sequential and hierarchical. That
is, students cannot move to the next higher level until they have successful
mastered the previous level. Furthermore movement from a lower level to the
next higher level depends more on content and pedagogy than on maturity and age.

The
van Hiele research shows students aren’t ready for formal proof until they
have had concrete experiences with geometric figures and have successfully
mastered earlier levels of visual thinking. Research shows 70% of high
school students enter geometry operating at level 0 or 1 on the van Hiele
measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989). Yet
"traditional" geometry textbooks that begin with establishing
postulates and proving theorems are expecting students to immediately begin
their geometry experience at levels 3-4. When the teacher and the textbook are
presenting geometry at van Hiele level 3 or higher, while the students are
functioning at level 0 or 1, it should be no surprise that there is such a high
failure rate in traditional geometry courses.

From
*Mathematician's Delight* by English
Mathematician W.W. Sawyer (1911-2008)

*It would, I suppose, be quite possible to teach a deaf and dumb
child to play the piano. When it played a wrong note, it would see the frown of
its teacher, and try again. But it would obviously have no idea of what it was
doing, or why anyone should devote hours to such an extraordinary exercise. It
would have learnt an imitation of music. And it would fear the piano exactly as
most students fear what is suppose to be mathematics.*

Some
geometry teachers claim they can successfully teach all of their geometry
students how to create geometry proofs. Perhaps they only have those 30% that
are entering geometry and functioning above van Hiele levels 0 and 1. I suspect
however, they may also be teaching an "imitation geometry." Their
students are trained to go through the motions of stating the theorem about to
be proved (5 points), then stating the given (5 points), then stating the show
(5 points), then drawing the diagram to the right (5 points), then drawing the
big T (5 points), then putting the given information in the first few lines of
the T-proof (5-15 points), then writing given to the right of each statement
(5-15 points), then throwing in enough statements to garner enough points to
get credit for the exercise without having any clue as to what he or she was
doing or why anyone should devote hours to such an extraordinary exercise. They
are doing imitation geometry.

*Discovering Geometry*, in its first edition, was an innovator
in addressing students’ needs for gradual development of the deductive process.
*Discovering Geometry* is the only high
school geometry textbook on the market that is aligned with the van Hiele model
and other research on geometric proof. We accept the vast majority of
student are entering geometry at very low van Hiele levels of geometric
reasoning and our goal, with careful deliberate scaffolding, is to move them to
higher and higher levels of geometric reasoning as they progress through the
course.

**The Role and Function of Proof **

Geometry
student's consistent difficulties with understanding proofs (an international
problem BTW) should not be solely attributed to their inability to reason but
perhaps our inability to recognize there are many purposes for doing
proofs and we have been stressing the wrong purposes for proof at inappropriate
times. Professor Michael de Villiers' research on the role and function of
proof identifies six basic roles for proof:

**• Verification
–**to remove doubt, to
convince someone of the truth of a statement

**• Systematization
**–organize known results
into a deductive system of postulates, definitions, and theorems

**• Explanation
**–insight into why
something is true

**• Discovery
**–proof can occasionally
lead to new unexpected results

**• Communication
**–proof can create a
forum for critical debate

**• Intellectual
Challenge **–proof can be
a testing ground for intellectual stamina and ingenuity

The
function of proof in a high school geometry course has been mostly two-fold: to
remove doubt, to convince someone of the truth of a statement –**verification** and to establish geometry
as a mathematical system –**systematization**.

Many
high school mathematics teachers seem to hold this naive view the main
function of proof is to provide verification that a given statement is true.
The role of proof as a means of verification is a useful method of verifying
the "truth" of a proposition within a mathematical system, especially
when coming across surprising (non-intuitive) results. However this view does
not stand up to actual mathematician's experiences. Professor George Polya
wrote, “When you have satisfied yourself that the theorem is true, you start
proving it.” If the sole or primary purpose for doing proofs in a high school
geometry course is to provide verification of the truth of a statement then
students functioning below level 4 on the van Hiele scale will continue to
question, or worse, disregard the process of proof.

If
systematization is emphasized as a primary function of proof right from the start
of a geometry course, the same poor results will persist. The van Hiele model
tells us systematization requires the highest van Hiele level of geometric
reasoning. Geometry textbooks that begin their geometry program with lists of
definitions and postulates, and then begin doing two-column proofs are ignoring
the research. Only an honors course or any class in which all students are
finally operating at van Hiele level 4 would have any chance of success in a
course that looks at geometry from the perspective of a mathematical system. Until
some magic happens and all students beginning geometry enter the course
functioning at van Hiele level 3, any attempt to create a mathematical system
for a regular or informal geometry course is likely to continue to have major
problems.

Students
can acquire a very high degree of confidence in a conjecture arrived at by
inductive methods but these methods may not explain why the conjecture is true.
Here is where proof can come to the rescue. An example would be the inductive
discovery that the sum of the measures of the three angles of a triangle is
always 180°. A good inductive first approach would be to ask students to
measure the three angles of a number of triangles thus gaining reasonable
confidence the sum is indeed 180°. The same can be done with dynamic
geometry software. Either investigative approach is a good first step because
it is the first approach students would take. These inquiry approaches do give
students confidence in their conjecture however they give no insight as to why
the sum is always 180°. The investigation should be followed by a second
investigation where they cut out the triangle and then tear off two of the
angles and arrange them on both sides of the third angle to create a straight
line. From this arrangement they can see the three angles create a line
parallel to the third side. This visually explains what properties this
conjecture is dependent upon and why the conjecture is true. This can also be
done quickly with a patty paper investigation.

From
*The Role and Function of Proof in
Mathematics* by Michael de Villiers:

* It is not a question
of "making sure," but rather a question of "explaining
why."*

**Explanation as the Primary Role of
Proof **

Using
proof as a means of explaining why something is true is the most meaningful
role proof can play in a high school geometry course. Asking why something is
true, after performing investigations that have convinced students it is
true, is a powerful 1-2 punch. Explaining why can be an effective tool
regardless of a student's van Hiele level. This is the approach we take with *Discovering Geometry*.

In
practically every lesson in the fourth edition of *Discovering Geometry* (DG4) students are asked to perform geometric
investigations and then make their geometry conjectures. After performing their
investigation and making their conjecture they are asked, "can you explain
why?" For example, after their very first two investigations leading to
geometric conjectures, the Linear Pair Conjecture and the Vertical Angles
Conjecture, students are asked:

"*Developing
Proof *You used inductive reasoning to discover both the Linear Pair
Conjecture and the Vertical Angles Conjecture. Are they related? If you accept
the Linear Pair Conjecture as true, can you use deductive reasoning to show
that the Vertical Angles Conjecture must be true?"

We
then ask them to work with their cooperative group members to develop a
paragraph proof explaining why the conjecture is true then check their
reasoning with ours.

Later
after discovering the Triangle Sum Conjecture students are asked:

"*Developing
Proof *The investigation may have convinced you that the Triangle Sum
Conjecture is true, but can you explain *why*
it is true for every triangle?"

We
then direct them to look back at the arrangement of the three angles they
tore off and reassembled forming a line. We ask, "how is the resulting
line related to the original triangle?" This is their lead-in to creating a
paragraph proof explaining why their conjecture fits with what they have
already discovered/proved about parallel lines.

The
inductive and deductive reasoning in DG4 continues with investigating,
conjecturing, and explaining why, from Chapter 2 through Chapter 12. It isn't
until the last chapter, when there may be students ready for van Hiele level 4
reasoning, that we introduce geometry as a formal mathematical system.

Again
from *Mathematician's Delight* by
English Mathematician W.W. Sawyer (1911-2008)

*The Great Pyramid was built in 3900 B.C. by rules based on
practical experience: Euclid's system did not appear until 3,600 *years later*. It is quite unfair to expect children to start studying
geometry in the form that Euclid gave it. One cannot leap 3,600 years of human
effort so lightly! The best way to learn geometry is to follow the road which
the human race originally followed: *Do* things, *make* things, *notice*
things, *arrange* things, and only then –*reason* about things. *

Once
deductive arguments began to sprout up around Ancient Greece, it still took awhile for the process to be accepted. From the first deductive arguments by
Thales to Euclid's Elements, it took over 300 years. We owe it to our students to
give them time to move up through the van Hiele levels so they come to
understand the role proof plays in mathematics.

* *